Décompositions de graphes et algorithmes efficaces

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Décompositions de graphes et algorithmes efficacesMichaël Rao

To cite this version:Michaël Rao. Décompositions de graphes et algorithmes efficaces. Autre [cs.OH]. Université PaulVerlaine - Metz, 2006. Français. NNT : 2006METZ007S. tel-01752468

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ë G2 = (V2, E2, lab2)8 ýZ> `òLG k @+ -Ãý ;h@ë ût

S ⊆ 1, . . . , k2G1×SG2

'þ ~`Ëò«LGE,-Ãý (V,E, lab) þB-Ãý V = V1∪V2h

E = E1 ∪ E2 ∪u, v : u ∈ V1, v ∈ V2

ë (lab1(u), lab2(v)) ∈ S

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lab1(u) u ∈ V1

lab2(u) u ∈ V2

gûT R : 1, . . . , k → 1, . . . , k G = (V,E, lab)

<jþ ûË ρR(G) = (V,E, lab′) þ-Ãý lab′(u) = R(lab(u)) hÝûËý ûËý u ∈ V

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GKî ç-/íB86M1¶çé

NLC-wd(G) ≤ cwd(G)íëï

cwd(G) ≤2× NLC-wd(G)

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GKî ç-/íO86M1¶çé

cwd(G) ≤ 3× 2twd(G)−1 gût G = (V,E) ý)`Ëò«LG i kâ Ø 4Þ ÚlÜ8ÚMÞ8KÜ8 8 G 5þ `ËòLG (V ∪E,R) hû

v, e ∈ R E v ∈ V h e ∈ E v ýa >K 98 e DÈþ û I(G) iaþ ? [ýL«ò

ûË -Kýaþò`Ë ýYò"ÝûË E2 98Fÿ ý `ËòLG[ þ òþò`Ë ýN892þM,-Ãý 8ý)`ËòLG98 ÿ 2;M8 L2ûÃ(8ýZ> òò. ' , þò' ð × × 2 2

cwd(I(G)) ≤ 3× 2twd(G)−1 íëï twd(G) ≤ 6× cwd(I(G))− 1

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eyi )^`PìD¤tPmftP [ìzg~t>[jkw :

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C = 〈V1, . . . , Vk〉[]_iKP`koiKprq`s>iKk,knmÌedq`silkN[ì

G moils¾^ìIg~t^`cÒmftd^jmv ∈ V |〈V ′ ∈ C : v ∈ V ′〉| ≥ w(v) µiÑR~Q¹X¡Y »X¡R~QS£j UVY ´ R~XTXÏ¡Ï[U]^`P¦acfedgjhì

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iKmw

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(G,w)ilknmr^`Pji ltds>tcoemftP

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Gim4^`Pì ¤tPmf>tdP©[jig~t[`k

w : V → N ()+3-¦`^jPì ltstcfemo>tPpvP`>pr^jp8[ì(G,w)

ÕY¸QSQSY JFLJ FL \ ^" 0 OFQR+Sh TVU U Qz] T a'aV] 8 YXW i W ]taV];a,432_Q ` Z Q f R+SR Q Um` Qba T YXR+W_]_^ ][S

f QR+S h1, . . . , hr f ] f f f R h ] f S TeUmU Qz] T a R U S][W U ]$ nOR

h] f S TeU ] ] T R ]_^Y QW f χw(Gh) = w(vh)

^) nOR

h] f S TeUmU Qz] T a ^ Y QW f χw(Gh) = maxi∈1,...,r χw(Ghi

)^

+ nORh] f S TeUmU Qz] T a ^/Y QW f χw(Gh) =

∑r

i=1 χw(Ghi)^

7% N V!,% ! "!$#%



V ∗ = v1, . . . , vr][S/S]& k T ] vi f QR+S ] f Q `;` ][SPW ]ZW4 f ] U SYXSR ta Tq` Qba T ] Vhi

OFQR+Sw∗ :

V ∗ → N TVU ] Q U 2[SR Q U aV] Z QRa f S]& ] k T ] w∗(vi) = χw(Ghi),l QW f χw(Gh) = χw∗(G∗

h)

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() -#- ^``[ìKpeP`[ì<g~t^`câ^`PÑacoedg`hìG

iKmâ^`Pk ∈ N

koG

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k iKkum !º¶Ktpwg`s>im\¨_° ³dª¬ Ë Pg~il^jmcuilpvedco^ìKc1^ìrmft^¸macfedgjhìltdP`Pì É ie\^`PÊiKP`koiKprq`si[jtpvPedPm KtP`Pì É i Õtd ls±eCyiKcoku>tP©[U]tgjmfpvkfemo>tPv[`^©g`cotqjs lpwi\1X®lY|Q¹ºW¤YTR~QZX£~ R~XXYÎµY !uR ´ ZQZ®l£j ¤R~9- 4 % -`^`P©acoedg`hì

G = (V,E) ()+3-)mÌeds>s>ixpvP`>pC^`p [U]_^jPwiKP`koiKprq`si4[jtpvPedPm KtP`Pì É i4[ìG +∞ kl]>s`P]_iKPvg~tkokl[ìgek

Ë PzP`tdmoiIgedcγ/(G)

s>emfed>ss>i<[U]^`Pzg`s^`kNgimfmxilPjkoilpCq`s>i[`tpw>PedPm ltdP`Pì É i1[ìG µiKkÐkot^jkn¶

g`cutq`s KpviKk\culkotd^`[`coi)ku^`crsà]_edcuq`coiz[jiz[`bKtpwgtkumo>tPpvt|[`^`s>ed>cuivkutPm\[`tP`Pjlk\gedcCs>izsilpwpviko^jyedPm ÕY¸QSQSYoJFLJ6FL OFQR+S

h TVU U Qz] T amaV] 8 YXW i W ] aV] a,432_Q ` Z Q f R+SR Q U ` Qba T YXR+W_]_^ ][S f QR+S h1, . . . , hrf ] f f f R h ] f S TeUmU Qz] T a R U S][W U ]$ nOR

h] f S TeU ] ] T R ]_^Y QW f γ (Gh) = 1

^) nOR

h] f S TeUmU Qz] T a ^ Y QW f γ (Gh) = +∞

^+ nOR

h] f S TVU U Qz] T a ^cY QW fCf8 R] 1 R f S] i ∈ 1, . . . , r S]&dk T ] Ghi

YXR+S TeUKf Q `;` ][SaVQ ` R U Y U S+^Y QW f γ (Gh) = 1

^ f R U Q U γ (Gh) = 2^

0 nORh] f S TVU U Qz] T a ^ Y QW f γ (Gh) = γ (G∗

h)^ Q

G∗h = (V ∗, E∗)

] f S ] WY[Z \ ]2$YXWY,2[S.4[W[R f SRk T ]saV] h 4 ` Q Uhf SWYXSNR Q U µilkwe *wcopvemftP`k im kutPmÑ>pwpvK[`±emfilk Ë P ku^`g`g~tkoi)[`tPhb|^ji

hilknm

tm D∗ ^`P ilP`kuilprq`sib[`tpw>PePmtltPjPì É i)[ìG∗

h Ë P KtP`knmfco^jm\s.]_iKP`koiKprq`s>iD

ilPg`cuilPedPm^`PkotdpvpwiKm1[ji

Vhi

gt^jcImft^jk1s>ilk1kutpwpvimfkvi

[edP`kD∗ F stcok

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2kotpwpvimfk D [`tpw>PìImot^jm4s>i1adcfedg`hji

Gh PyilcukoiKpviKPm kotm D iKkum1^jP¼ilP`kuilprqjs>ir[`tdpvPedPmI[ji

Gh iKm%kotmD∗ s.]_iKP`koiKprq`siC[ìKk

vi

mfils^ì

D ∩ Vhi6= ∅ stcok

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gedcD ku

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kmoils^ì

vj, vk ∈ E(G∗h)

iKmD ∩ Vhk

6= ∅ t^jmÐkotpwpvimÐ[ìG∗

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kl]>sPU]_iKPÊg~tkuk$l[jigek Ë PSP`tdmoivgèdc

γ /5Ù (G)s±ezmÌeds>sir[ìws±ebg`s>^jkg~iKmomfi Ks>>^ìÑ[`tdpvPedPmfi\[ì

G Ë PeÊsiÑs>ilpwpwiko^jyedPm s>erg`coiK^jyiilknmku>pw>s>ed>cuiI\s±e\g`cuil^jydigjcobKl[ìKPmfi ÕY¸QSQSYoJFLv 2L OFQR+S

h TVU U Qz] T amaV] 8 YXW i W ] aV] a,432_Q ` Z Q f R+SR Q U ` Qba T YXR+W_]_^ ][S f QR+S h1, . . . , hrf ] f f f R h ] f S TeUmU Qz] T a R U S][W U ]$ nOR

h] f S TeU ] ] T R ]_^Y QW f γ (Gh) = 1

^) nOR

h] f S TeUmU Qz] T a ^ Y QW f γ (Gh) = +∞

^+ nOR

h] f S TVU U Qz] T a ^cY QW fCf8 R] 1 R f S] i ∈ 1, . . . , r S]&dk T ] Ghi

YXR+S TeUKf Q `;` ][SaVQ ` R U Y U S+^Y QW f γ (Gh) = 1

^ f R U Q U γ (Gh) = 2^

0 nORh] f S TeUmU Qz] T a ^ Y QW f γ (Gh) = γ (G∗

h)^/Q

G∗h = (V ∗, E∗)

] f S ] WY[Z \ ]2$YXWY,2[S.4[W[R f SRk T ]saV] h 1X®lY|Q¹ºW¤Y® £`¹@WZYTXR~QZX£` !uR ´ ZQZ®l£j ¤R~9- 4 % -`^`P©acoedg`hì

G = (V,E) ()+3-¦jmfed>ss>ipvP`>pr^jp []_^`P©ilPjkoilpCq`s>ikumfedq`si[jtpvPedPm<[ìG

Ë PP`tdmfiCgedcγ Ü (G)

s>ewmÌe>s>si[U]_^jP¦gjs>^`kgimfmIilP`kuilprqjs>ir[`tdpvPedPm KtP`Pì É iw[jiG µirg`cut¶

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G = (V,E)im4^`Pì ¤tPmf>tdP©[jig~t[`k

wko^`c

V ()+3-¦`g~t[`k<pvP`>pC^`pγ Ü w(G)

[U]^`PÊiKP`koiKprq`siknmÌedqjs>i1[`tpw>PèdPm[ìG

ÕY¸QSQSYoJFLvM L OFQR+Sh TVU U Qz] T amaV] 8 YXW i W ] aV] a,432_Q ` Z Q f R+SR Q U ` Qba T YXR+W_]_^ ][S f QR+S h1, . . . , hrf ] f f f R h ] f S TeUmU Qz] T a R U S][W U ]

$ nORh] f S TeU ] ] T R ]_^Y QW f γ w(Gh) = w(vh)

^) nOR

h] f S TeUmU Qz] T a ^ Y QW f γ w(Gh) =

∑r

i=1 γ w(Ghi)^

+ nORh] f S TeUmU Qz] T a ^/Y QW f γ w(Gh) = mini∈1,...,r γ w(Ghi

)^



V ∗ = v1, . . . , vr][S/S]& k T ] vi f QR+S ] f Q `;` ][SPW ]ZW4 f ] U SYXSR ta Tq` Qba T ] Vhi

OFQR+Sw∗ :

V ∗ → N TVU ]Q U 2[SNR Q U aV]Z QRa f S]& ] k T ] w∗(vi) = γ w(Ghi)Vl QW f γ w(Gh) = γ w∗(G∗

h)

!@t^jc P`c tP[`tPjPìs.]edsatcumfhjpvibg~t^`c - 45- *,$3- () tm G = (V,E)^`P

acoedg`hì f : V → 1, 2^`PÊmf|^jiKmÌeai1[ìlk4kotdpvpwiKmok<[ì

VimD ⊆ V tm

a(v) =

2kuf(v) = 2

imvilknm4^`PÊkutpwpvim>kotds>%[ì

G[D] 1

ku>P`tP

7% N V!,% ! "!$#%

w(f,G,D) =∑

v∈D

a(v),

γ(f,G) = minD⊆V

w(f,G,D)t#

DilknmI^jP©iKP`koiKprq`siknmÌedqjs>i[ì

G ÕY¸QSQSYoJFLvJFL OFQR+S

h TVU U Qz] T amaV] 8 YXW i W ] aV] a,432_Q ` Z Q f R+SR Q U ` Qba T YXR+W_]_^ ][S f QR+S h1, . . . , hrf ] f f f R h ] f S TeUmU Qz] T a R U S][W U ]= OR

h] f S TeU ] ] T R ]_^Y QW f γ(Gh) = 1

= OR

h] f S TeUmU Qz] T a R U S][W U ] 4[SRk T ][S 4 ^ Y QW f γ(Gh) =

∑r

i=1 γ(Ghi)

= ORh] f S TeUKU Qz] T aR U S][W U ] 4[SRk T ][S.4 ^ Y QW fCf8 R] 1 R f S] TeU i ∈ 1, . . . , r S]&dk T ]

Ghi

YXR+S TVUqf Q `;` ][SPaVQ ` R U Y U S+^ Y QW f γ(Gh) = 1^ f R U Q U γ(Gh) = 2

= OR

h] f S TeU U Qz] T a R U S][W U ] 4[SNRk T ][S.4 cOFQR+S V (G∗

h) = v1, . . . , vr^t][S f QR+S f 8 4[SRk T ][S Y ] aV] fCf Q `;` ][S f aV] G∗

h

S]& k T ] f(vi) = 1 f R Ghi

Y TVUKf Q `;` ][S aVQ ` R U Y U S^ 2f R U Q U Fl QW f γ(Gh) = γ(f,G∗

h)

4 ` Q Uhf SWYXSNR Q U µilktdq`koiKcuyemftP`kwgtd^`cÑs>ilkwP`t|il^`[jkKmo>^ìmflk ¦iKm SkutPm>pwpwl[`>emfiKk tm h ^`P P`t|il^j[ tm D∗ ^`P kut^`k5¶3ilPjkoilpCq`s>i¼[jtpvPedPmv[ì

G∗h

|^jpw>P`pvkoi

w(F,G∗h, D

∗) Ë PjKtP`knmfco^jmÒ^`PÑkut^`kn¶DilP`kuilprqjs>iD

[ìV (Gh)

[ì<s±eIpvedP`lcuiÐko^`yedPmfi vi ∈ D∗ iKm

f(vi) = 1 edstcoktP eutd^jmfiD

^`P kutpwpvimC[`tdpvPedPm[ìGhi

vi ∈ D∗ f(vi) = 2

imvi

e^`Pydt>ku>P¼[eP`kG∗

h[D∗] eds>tdcok4tPeut^jmoiC^`PkutpwpvimI[ì

Ghi

D È P Pku

vi ∈ D∗ f(vi) = 2iKmvi

P]eÑgedk<[ì1ydt>ku>P)[edP`kG∗

h[D∗] tdPÊe ut^jmfi1^jPÊkutpvpwiKm[ji

Ghi

im4^`PkotdpvpwiKm1[ji

Ghj

D t# j ∈ 1, . . . , r moilsÒ^ì

vi

iKmvj

kotilPmed[oe lilPmok[èdP`kG∗

h tdP¦koedmÛ]^`PmfiKs

ji É >kumoi edc4s>iacfeg`hìilkumIg`coiKpvilc silkum ZeXls>i1[ìytc^ì

|D| = w(f,G∗h, D

∗) iKm4^ìD

ilkum4^`P©ilPjkoilpCq`s>i[`tpw>PèdPm<[ìGh tP

γ(Gh) ≤ γ(f,G∗h) tm

D^`PÑilPjkoilpCq`s>i<[`tdpvPedPmºpw>Pj>pr^`p¯[ji

Gh tm i ∈ 1, . . . , r moilsj^ì D∩V (Ghi) 6=

∅ Ghi

ew^jPkotdpvpwiKm[`tdpvPedPm td^ÊkK]_sµi É >knmfij 6= i

mfils^ìD ∩ V (Ghj

) 6= ∅iKmvi, vj ∈

E(G∗h) edstcok |D ∩ V (Ghi

)| = 1 ko>PjtPtPg~t^`cocoedm ltPjkumfcu^`>cui\^`PilP`kuilprqjs>i\[`tpw>PèdPm%g`s^`kg~iKmfm^ì

D >P`tP |D ∩ V (Ghi)| = 2 tdP

w(f,G∗h, D

∗) = |D| t# D∗ iKkum4sà]ilPjkoilpCq`s>i1[ìKkvi g~t^`c<s>iKko^ìKs>k

D ∩ V (Ghi) 6= ∅ tP

γ(f,G∗h) ≤ γ(Gh)

µig`cutq`slpwi4Û]>s Zed^¸mxculkut^`[`cuiIko^jc,s>iKkÕacfedgjhìlkxg`coiKpvilcokÕiKkumx[`tP^`Pì4alP`Kcfeds>koemftPÑ[`^g`cutq`s Kpvi ()#() pved>k>s@Pì\kl]_edam1gedk1[ì\koezyilcukotPgtPj[`lculi Ë Pg~il^jmcuilpvedco^ìKc^ì1[edPjk li ek tPbP]_iKkumgedk<tdq`s>aI[ì edsl^`silcÐsilk,yedsil^`cukÐgtd^`c,mftd^`k<s>iKkÐP`t|il^`[`k<[ji1sà]_edcoq`cui È PbiKiKm kos>i%acoedg`hìIilkum/ltP`Pji É i silkÐkuil^`silk<>Pe¤tcopvemftP`kÕP` lilkukfedcoiKk<gt^jc/edsl^`silcxs±emfed>ss>i[U]^`P\[ìKkâg`s>^jkºg~iKmfmfk@ilP`kuilprq`silkâ[jtpvPedPmfkXkotdPmâs>ilk@>Ph¤tdcopvemftP`k[ì,s>ecoeXlPì sà]Kmf^ìKmumfi,imÒs>iacoedg`hì ledcfeXmfKco>knmf|^ji iKm gt^jc/Ìhed^ì sk<[ì1s±eCcoeXlPì [ì1kfeytd>c<kos>i%kut^`kn¶Dacfeg`hì1>Pj[`^`mgedcsi1pvt|[`^`si ltpwgtdcumfi1^jPÊkutpvpwiKm4[`tdpvPedPm ^`g~il^jmpmfcoimfcut^jydiKP©moilpwg`k

O(n+m) s>i©acfedgjhì¼PU]ilknmbgedkDKtP`Pì É i s Ze^jmDleds K^`s>iKcvs>e¦mÌe>s>si)[U]_^jP [ìlkzgjs>^`kvgimfmokiKP`koiKprq`s>iKk[`tpw>PèdPmfk<[ì Ìhed^ìnltdpvg~tkfePmoi ltPjPì É i

"!$#%

/10 10 %¼MÕ= àE.EN91:

C»XY|QZAZXT@VX¤ 4 % -`^`P©acoedg`hì

G = (V,E) - 4 ()A`^ìKsilkum4si1g`s>^`k<adcfedP`[kmfiKsµ^ì

Gedm<^jP

Pk

P`[`^`mÍ i)g`cotqjs lpwiziKkum !º¶Ktpwg`s>im cu ¤KcoilPhli¨° ¬,[edP`k©¨_° ³dª¬ Ë P P`tdmfi

ip(G)sizg`s^`k

acoedP`[kmfilsµ^ì

Gedm<^`P

Pk

>Pj[`^`m ÕY¸QSQSYoJFLvvFL OFQR+S


h] f S TeU ] ] T R ]_^Y QW f ip(Gh) = 1

= OR

h] f S TeUmU Qz] T a R U S][W U ] 4[SRk T ][S 4 ^ Y QW f ip(Gh) = maxi∈1,...,r ip(Ghi

)

= ORh] f S TVUqU Qz] T a R U S][W U ]64[SNRk T ][S.4 ^Y QW fcf RZ Q T W SQ T S i ∈ 1, . . . , r ^ G(hi)

] f S2_Q ` Z ][S+^Y QW f ip(Gh) = 2^ f R U Q U ip(Gh) = max

(3, ip(Gh1), . . . , ip(Ghr

))

= ORh] f S TeUmU Qz] T a R U S][W U ] 4[SRk T ][S 4 ^ Y QW f

ip(Gh) = max(3, ip(G∗

h), ip(Gh1), . . . , ip(Ghr)).

egjcoil^¸yi4ilkumÕpvpwl[j±emoi edcÒsi<acfeg`hìPk g~t^`c k ≥ 4 ilkumÕgjcoilpw>iKc iKmN[`tP<Pji<giK^jmNgedk

Kmocoi1[` ltpwg~tko1gecÐs±er[` ltpwg~tkomf>tdPzpvt|[`^`s>ed>cui[`^)acfedgjhì µi1silpwpviP`t^`k<[`tdP`Pì1[`coibmfi ¶pwilPm^`PedsatcumfhjpvicobK^`coku,gt^jcnleds K^`s>iKc

ip(G)iKP^¸mf>s>koedPm1s±eb[` ltpwgtdkomo>tPpwt¸[j^`s±edcoi

[ìG

si1g`cotdq`s Kpvi\coKkot^j[`coiko^`csilk<acoedg`hìKkpledcfe KmfKcokumf^ìlkilknm4s>i1g`cutq`s Kpvi1tdco>ad>Peds /10 10 N=@=â9 &NJ º=xH = #?¸=A

tmG = (V,E)

^`Pvacfeg`hì OPvilP`kuilprqjs>iF ⊆ V

iKkumÐ^jP gtd^`c ]_]$a i Y,2 X][W8S] 1;f ][S kl]>s?ltPmfilPmed^©pvtP`kÐ^`P©kotdpvpwiKm<[jinÌhèd^ìsÓVls>i[ìG

µYY¸T¹X£ ¢~Y|¡ YÎ ®lY 4 % -`^`P©acoedg`hì

G = (V,E) - 4 ()A`^ìKs>si1ilkum4s>e\mfed>ss>i[U]_^jP¥ [ì1mÌeds>s>ipvP`>pC^`pG[ìG

Í iÑg`cotqjs lpwiriKkum !º¶Ktpwg`s>im cu ¤lcuilPKi©¨° ³¬Ò[eP`kw¨° ³dª¬ Ë PP`tmfilcoeµ(G)

s±emfed>ss>i[U]^`PÑg`s>^jkÒgimfmÒ¥ [ì

G O4PÑilP`kuilprq`siV ′ ilknmÕ^`Pì R~V¢~Y¡ V¡YvTY®lR~QQY ®,ko`g~t^`câmot^jmoi

edcKmfiu, v ∈ E u ∈ V ′ t^

v ∈ V ′ Ë PP`tdmoiβ(G)

s±e)mÌeds>s>i\pw>Pj>pr^`p [U]^`PìDKt^jyiKcumo^`coi[ìkotpwpvimfk[ì

G sXilknmcltdP`P^¼^ìβ(G) = n − α(G) ytcgèdci É ilpwg`s>iv¨° ³dª¬ Í edsl^`silc

β(G)coiy|>ilPm4[`tPh;leds K^`s>iKc

α(G) tmG = G′ G1 ... Gr

v1 ... vr eydib G′ = (V ′, E ′)

iKmgtd^`c1mft^jmi ∈ 1, . . . , r Gi = (Vi, Ei) imkotdm

f : V ′ → 1, 2, 3, 4^`Pbmf|^jiKmÌeai%[ìKkÐkotpwpwiKmfkx[ì

G′ edP`k/KiKmumfi1koi Kmo>tPbtP)eg`giKs>silcfeW¤Y ´ R~.TX®1[ì

fs±eryeds>iK^`c<ko^`yedPmfi\

w(f,G′, G1, . . . , Gr) =r∑

i=1

a(f(vi), Gi)

7% N V!,% ! "!$#%

t#G = (V,E)

im

a(e, G) =

µ(G)ko

e = 1,

β(G)ko

e = 2,

|V | − 1ko

e = 3,

|V |ko

e = 4

Ë PÊ[`mÛ]_^jPÊmf>^ìmÌedadif

iKkum¢`£`W.ZTXYvkl]>sUcuilkugi Kmfis>iKk ltP`[jmftP`kÐku^`yedPmoilkÇ s.]_iKP`koiKprq`s>i1[jilk4kotpwpvimfk<mf>^ìmfKk

4¤tcupviKPm^`P©¥ [ì

G′ Ç kuµ^`PÊkutpwpvim<ilknmImf>^ìmf1 eds>tcukÐmft^jk4kuilk<ytd>koP`kkotPm4Kmo>^ìmflk

4 Ç kuº^`PkutpvpwiKmIilknmKmf^ìKmo2 eds>tcukmot^`k%kuilk%ytd>koP`kIkutPmKmo>^ìKmolk

4koed^hxed^g`s^`k%^`P ^`µiKkum4Kmo>^ìKmo

3 Ë PÊP`tmfiÑµ′(G′, G1, . . . , Gr) = min

fw(f,G′, G1, . . . , Gr)

t#f

ilknmI^`P)mf>^ìmÌedadiyeds>[ìI[ìG′.

ÕY¸QSQSYoJFLvL OFQR+Sh TVU U Qz] T amaV] 8 YXW i W ] aV] a,432_Q ` Z Q f R+SR Q U ` Qba T YXR+W_]_^ ][S f QR+S h1, . . . , hrf ] f f f R h ] f S TeUmU Qz] T a R U S][W U ]

= ORh] f S TeU ] ] T R ]_^Y QW f µ(Gh) = 0

= OR

h] f S TeUmU Qz] T a R U S][W U ] 4[SRk T ][S 4 ^ Y QW f µ(Gh) =

∑r

i=1 µ(Ghi)

= ORh] f S TeUmU Qz] T a R U S][W U ] 4[SRk T ][S 4 ^/Y QW f

µ(Gh) = mini∈1,2,...,r

(min(µ(Ghi), β(Ghi

)− 1) + n(Gh)− n(Ghi)).

= ORh] f S TeUmU Qz] T a R U S][W U ] 4[SRk T ][S 4 ^ Y QW f

µ(Gh) = µ′(G∗h, Gh1, . . . , Ghr

). 4 ` Q Uhf SWYXSNR Q U tm f ^`P Kmo>^ìKmfedaibyeds>[ì tm gtd^`c\mft^jm

i ∈ 1, . . . , r Fi

^`P¥ pw>P`pr^`p [ì

Ghi Ci

^`Pji Kt^jyiKcumo^`coi[ìkutpwpvimfk)pvP`>pC^`p [jiGhi

V ′i ⊂ Vhi

moils^ì|V ′

i | = |Vhi| − 1 iKm Vi = Vhi

Ë P)[` Pjm

Ai =

Fi

kuf(vi) = 1,

Ci

kuf(vi) = 2,

V ′i

kuf(vi) = 3,

Vi

kuf(vi) = 4,

iKmF =

⋃ri=1Ai.

Ë P©g~il^¸mcuilpvedco^ìKc|^ji|F | = w(f,G∗

h, Gh1, . . . , Ghr) Ë PpwtPmfcuiN^ì

FiKkum@^`PC¥ [ì

Gh tdm C = (u1, u2, . . . , uk)^`PsÓVls>iN[ì

Gh ^`g`g~tkotdP`k^ì\ed^hl^`Pui

PU]eg`gedcnmf>iKPm1F Ë P[` Pjm

u′1, u′2, . . . , u

′k

moilsX^ìrg~t^`cmot^jmj ∈ 1, . . . , k

uj ∈ Vi

koµimkuil^`silpwilPm4kou′j = vi Ç xmft^`kÑsilk

vj

kotdPmw[`KcoiKPmok edstcok C ′ = (u′1, . . . , uk)ilknmw^`PoKÓVlsi©[ji

G∗h Ë crP`td^`k

koe ydtP`k1^ì\s.]_iKP`koiKprq`s>iC[ìlk1kutpvpwiKmokImf>^ìmfKk4¤tcopwilPmI^`P¥ [ji

G∗h tP\sXÓe

ed^pvtP`k,^jPbkotpwpvim,[jÊKÓVlsiC ′ |^jilknmKmo>^ìKmo

4 sÓeC[`tdP1ed^bpwt>Pjkx^`Pbkutpwpvim[ì

C[èdP`k

F ~Í tPmfcfe[` mf>tdP

"!$#%

Ç P`tPrs¸i É >kumoiimfiKsj^ì

vi

edg`gèdcfemÒed^\pwtP`k@[jil^ É ¤t>k@[edP`k(u′1, . . . , u

′k) tdP

f(vi) ∈1, 2 Ç ko

f(vi) = 1edstcokxmft^`k<silkÐydt>ku>P`k<[ì

vi

kotPm4Kmo>^ìmflk4 Í tpwpvi%ed^l^jPÊkutpvpwiKm,[ì

CPU]_iKkum[edP`k

F mft^jk4silku′j j ∈ 1, 2, . . . , k kutPmIKaed^ É tPh

C ⊆ Vi Ë cF ∩ ViilknmI^`PÊ¥ [ji

Ghi [U]t#'ltPmfcoed[`KmftP Ç koP`tP

f(vi) = 2 s¸ÓCe%ed^\g`s^`kº^`PÑkutpvpwiKm@[edPjkÒs>ilk@ilP`kuilprqjs>ilkÒhjtpvtda lPìKkâpve É >pved^ ÉltcucoiKkog~tP`[edPmok¼ed^ É ytko>Pjk©[jivi

|^jP]_iKkumÊgedk©[edP`kF s>tcukbkotmbmft^jkÊsilk

u′j j ∈ 1, 2, . . . , k

kutPmblaed^ É imbtP e^`Pì=KtPmfcfe[` mf>tdP ledc^jPì=ltd^jyiKcumf^jcoi¼[ìkotpwpwiKmilkum^`P¥ kutm4^`PiKm^`Pkuil^`s@kotpwpvim4[ì

C ′ iKkumI[`lcuilPmI[jivi tP>s

i É >knmfiKcfedmC^`P Ìhìlpw>P[ìvmfed>ss>irku^`g~lcoil^`cuit^laes>iw)[edP`kGhi

^`NPìvgedkokuigeced^K^`Pkotpwpvim%[ì

F µÍ tdPmocfed[j Kmo>tPeyibÑsisZedm%^ìF ∩ Vi

iKkum^`Pji;lt^jydilcnmf^`cui\[ìkotpwpwiKm[ì

Ghi

tPhF

ilknmÑ^`P ¥ [ìGh iKm µ(Gh) ≤ |F | âÍ tpvpwi Kils>e¼ilkumCy¸coedÐg~t^`cCmot^jm\mf|^jiKmÌeai

yeds>[ì µ(Gh) ≤ µ(G∗h, Gh1, . . . , Ghr

) tmF

^`P)¥ [ìG = G∗

hGh1

... Ghrv1 ... vr

h s>tcuk,g~t^`cxmot^jmi F ∩ Vi

iKkum<^jP)¥ [jiGhi

tmf : V ∗

h → 1, 2, 3, 4s.]_mf|^jiKmÌeai1ko^`yedPm

f(vi) =

1koF ∩ Vi

PU]ilkumgedk^`PjinKt^jydilcumo^`coi[ìkotdpvpwiKmok<[ìGhi

2

koF ∩ Vi

ilknmI^`PjinKt^jydilcumo^`coi[ìkotdpvpwiKmok<[ìGhi

im|F ∩ Vi| < |Vi| − 1

3ko|F ∩ Vi| = |Vi| − 1

4koF ∩ Vi = Vi.

Ë P coilpvedcu|^ji©^ìw(f,G∗

h, Gh1, . . . , Ghr) ≤ |F | ledc µ(G) ≤ β(G) ≤ |V | − 1

g~t^`crmftd^jmacoedg`hì

G = (V,E) Ò]_iKP`koiKprq`s>iI[ìlkÐkutpwpvimfk,mf|^jiKmfKk4¤tdcopwi^`P)¥ [ì

G∗h koP`tPztPgt^jcocfemltP`knmfcu^`>cui

^`PEÓeKs>i%[jiGh

rgecumfcx[]_^`P'ÓeKs>iC ′ [ì G∗

h

ilPbgjcoilPèdPm<^`PbkotdpvpwiKm,P`tPEltd^jyiKcumgedcF

gecilPjkoilpCq`s>ih`tpwta KPì ltdcocoiKkog~tP`[ePmI\^`P©kotpwpvim<[`^rKÓVlsi

C ′ tmimfilsX^ì

F ∩ Vi

Pjirkotm4gedk%^`Pjitltd^jyiKcumf^jcoir[ìkotpwpwiKmfk[ìGhi

es>tcuk4g~t^`c4mft^¸mjmfiKsÕ|^ji

vi

imvj

kotd>ilPmCeu[eXKilPmfk Vj ⊆ F koP`tP¦>sÒi É kumfiKcfedm^`PKÓeKs>iw[ìÑmÌeds>s>i3[`tdPms>iKk

kotdpvpwiKmok4PjikutPmIgek[eP`kF tPhku

f(vi) = 1 eds>tdcok<mft^jk4silkytkoP`k[ìvi

kutPmImf|^jiKmfKk4 tm

imfils|^ji

|F ∩ Vi| < |Vi| − 1 eds>tdcokÐsUPì1g~il^jmgekÓzeytc<[ìl^ É kotpwpvimfkÐgèdcopwmftd^`ks>iKkÐkotdpvpwiKmokÐ[ìlk<pwt|[`^`s>iKk/ltdcocoiKkog~tP`[ePmoked^ É ytko>PjkÐ[ì

vi

^`µPì%kut>iKPmgèdkÐ[edPjkF ko>PjtPtPvgt^jcocfemdltdP`kumoco^`coi4^`P ÓeKs>i4[ìmfed>ss>i

4[`tdPmxs>iKkxkutpwpvimfkÕPìIkutPm,gèdkx[èdP`k

F tPhmftd^`ks>iKkÐytkoP`k[ì

vi

kotdPm4mf>^ìmfKk4gec

f kfe^hÒed^©g`s^`k<^`P©|^jµilkum4mf>^ìmf3 Ò]_mf>^ìmÌedadi

fiKkum\yeds>[ì [`tdP w(f,G∗

h, Gh1, . . . , Ghr) ≥ µ′(G∗

h, Gh1, . . . , Ghr) iKmÑgec

ltdP`koK|^jilPKi|F | ≥ µ′(G∗

h, Gh1, . . . , Ghr) Í tpwpvinliKs±eviKkumy¸coedXgt^jc4mot^jm1¥ [ji

Gh tPeµ(Gh) ≥ µ′(G∗

h, Gh1, . . . , Ghr) Ë PÊe PedsilpwilPm

µ(Gh) = µ′(G∗h, Gh1 , . . . , Ghr

) Ë PÊg~il^jmpKtPls^`coi\Ç

hilknmI^`Pji ¤iK^`>ss>i edstcok Gh

ilkum4^`P©kotdpvpwiKmkots im µ(Gh) = 0

7% N V!,% ! "!$#% .

Ç hiKkum<mf|^jiKmf eds>tdcok G∗

h

ilknm<^`Pbacoedg`hìIkfedP`kec$Kmoilk sà]Kmo>^ìmÌedai f : V ∈V ∗

h → 1ilkumyeds>[ì iKm4kutPÊg~t[`k<ilknmpw>Pj>pr^`p Ç

hiKkumKmo>^ìmf edstcok G∗

h

iKkum^`Pbacoedg`hì Ktpwg`s>im OPbKmo>^ìmÌedai%yeds>[ì4[`tPjPì^`Pjiyedsil^`cÒ[jlcoiKPmfi[ì

4gtd^`cNed^Ñg`s>^`k %kotdpvpwiKmok >P`ku kutmÕ^`PÑkutpvpwiKmÕiKkumNKmo>^ìmf

1iKm4mft^jkIs>iKkIe^jmfcuilkIkutPm%mf>^ìmfKk

4 kotdm4^`P¼kutpvpwiKm4iKkum%mf|^jiKmf2 ^`Pe^jmfcui

3iKm%si

cuilkumoi4 µilk4ed^jmocoiKk4mf|^jiKmÌeailk<yeds>[ìlk<tdPm^jPÊg~t>[jk<ko^`g~lcu>il^jc4td^ÊKaeds Ç

hilknmImf>^ìmf tdPe\qìlP

µ(Gh) = µ′(G∗h, Gh1, . . . , Ghr

)

/10 10 7 &xHK=9= ? NLxJÕHKJ=x:<Hl? .E (¯J: tm

G = (V,E)^`P acoedg`hì imbkotm

u, v ∈ Vmfilskz^ì

uiKmv

kotd>ilPmb[èdP`kbs±ep Kpviltdpvg~tkfePmoi;ltPjPì É iw[ì

G O4Pkut^`k5¶3ilPjkoilpCq`s>iS ⊆ V \ u, v

ilknm^`Puv

®lÏ ´ £`¡£j Y|VX¡bkuuiKm

vkotdPm[èdP`k<[ìK^ É Ktpvg~tkoedPmfilk<[jlcoiKPmfilk<[ji

G− S S ilknm^`Puv

®lÏ ´ £j¡£¸ Y|VX¡vQZ@ZQS£WkoS

ilknm\^`Puv

¶3kulgecfemoil^`ciKmrg~t^`cmft^¸mS ′ ( S S ′ PU]ilknm\gedk^`P

uv¶3kulgedcoemfiK^`c S ilknm\^`P

®lÏ ´ £`¡£j Y|VX¡Q.XZQS£WÐkl]>sxi É >knmfiu, v ∈ V

moilsÐ^ìS

kotmr^`Puv

¶3kulgecfemoil^`crpw>Pj>pveds Ë PcoiKpeco^ì^ì

∅iKkum4^`P©kolgèdcfemoil^`cpw>P`pes~kuµiKm4koiK^`s>iKpviKPmkuµs>i1acoedg`hìPU]ilknm4gedk ltPjPì É i tm

s(G)s>i1P`tpCq`coi1[ìkoKgedcoemfiK^`cokpw>P`pe^ É [ì

G ÕY¸QSQSYoJFLv FL OFQR+S


h] f S TeU ] ] T R ]_^Y QW f s(Gh) = 0

= OR

h] f S TeUmU Qz] T a R U S][W U ] 4[SRk T ][S 4 ^ Y QW f s(Gh) = 1 +

∑ri=1 s(Ghi

)

= ORh] f S TeUmU Qz] T a R U S][W U ] 4[SRk T ][S 4 ^/Y QW f s(Gh) =

∑ri=1 s(Ghi

)

= ORh] f S TVUEU Qz] T a R U S][W U ]4[SRk T ][S.4 ^Y QW f s(Gh) = s(G∗)+

∑r

i=1(s(Ghi)−ai)

^Qai = 1 f R Ghi

U*8 ] f ShZhY f 2_Q U U ] 1 ] ][S NG∗h(vi)

] f S TVUjf 4ZhYXWYXS] T W ` R U R ` YhaV] G∗h

âi = 0

f R U Q U 4 ` Q Uhf SWYXSNR Q U h ilknmbKmo>^ìmf es>tcuk

∅iKkum)^`Pkulgecfemoil^`cbpw>P`pes imbs>iKk

ed^jmocoiKkNkulgecfemoil^`cukxpw>P`pe^ É kotPmN[ìlkÕkulgedcoemfiK^`cokNpw>P`ped^ É [jilkGhi

µilkNacoedg`hìKk Ghi

kutPmltdP`Pì É iKk [`tPPU]tPm4gedk

∅ltpwpviIkolgèdcfemoil^`c tm

hiKkumIKmf^ìKmo >sµi É kumoiC^`P

uv¶DkoKgedcfemfil^jc4pw>Pj>pveds

S edstcok u Pì[`tmgèdkKmocoiÑed[oeXliKPm

v iKm1[`tdP\>sXi É >knmfii ∈ 1, . . . , r

moilsº^ìu, v ∈ Vhi

edP`k li;ledkVhj⊆ Sg~t^`câmftd^jm

j 6= i iKm S∩Vhi

iKkumN^`Puv

¶3koKgedcoemfiK^`cÒpvP`>pveds[edPjkGhi

X >Pjko Ìhed^ì4koKgedcoemfiK^`cpw>P`peds¾[ì

Gh

Ktcocuilkog~tP`[Ê\^`P©kolgèdcfemoil^`cpw>P`pes[U]^`PGhi

È P Pkutmh

Kmo>^ìKmo tm u, v ∈ Vh sUÓ)e\[ìK^ É ekrmocfedmfilc Í edk ¸>s~i É >knmfi%^`P

i ∈ 1, . . . , rmfiKs^ì

u, v ∈ Vhi O4P

uv¶3koKgedcoemfiK^`c,pw>P`pes

S[jiKy|cfe

¤tcllpwilPm>PKs>^`cui\mft^`ks>iKkVhj

eydib vi, vj ∈ E(G∗h)

imCg~t^`c1mot^jmj 6= i S ∩ Vhj

6= ∅ku

iKmkoiK^`s>iKpviKPm%kovi, vj ∈ E(G∗

h) ecI>sµiKkumIpvP`>pveds ig`s>^jk ko u iKmvkotdPm[edPjkIs±eÑpjlpwi

ltdpvg~tkfePmoi ltP`Pji É i[jiGhi

edstcok S ∩ Vhi

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Gg~il^jm Kmocoi^jmfs>>ku1gtd^`c edsl^`silc<s>ig~t>[jk4pve É ¶pr^`p#[]_^`Pji Ks>>^ìku^`c,^`Pvacfeg`hì lt¶Ìh`tcu[eds

GilPwmfilpwg`k

O(n2) Ë Pvs>ipvt|[` i<gtd^`cxg~t^jydt>ces l^js>ilc

ωw(G)iKP©moilpwg`ks>>Pjedcoi Ë P coedg`g~ils>sibsà]_eds>adtcomfh`pwiv[ji¥cfeP s ltpwpviKPli)gec;edsl^`silcÑ^`P'tco[`cui)[U]_Ks>pvPemftP

gedc"Zedmv1, . . . vn

[`^Sacfedgjhì[U]ilPmocoliG = (V,E) eyi w(vi) ≥ 0

s>iwg~t>[`k[ìvi

g~t^`ci ∈

1, . . . n OPbiKP`koiKprq`s>ikumfedq`s>i[ìIg~t[`k,pve É >pr^jpilknm ltPjkumfcu^`m,[ìIs>enZe0KtPzku^`yedPmfi Pjmf>eds>iKpviKPm kotdmc w(vi) = w(vi) g~t^`c<mft^¸m

i ∈ 1, . . . , n !Xt^`c<mft^¸mi[ì

1n ko c w(vi) > 0 es>tcuktP ltds>tcu>i

vi

ilPwcot^àdi iKmNtPÑkut^`kumocfedmc w(vi)

c w(vj)

g~t^`câmot^jmvj ∈ N [vi]∩vi, . . . vn & êdP`[bmot^`ks>iKk<kotpwpvimfkÐtdPm4mf%mfcoedmolk kotm I = ∅

im g~t^`cÐmot^jmi[ì

n

1 ku vi

ilknm4cot^àdiiKmPjtP)ed[oeXKilPm4\ed^K^`P)kotpwpwiKmÐ[ji

I eds>tdcok I = I ∪ vi & êdP`[zmot^`k<silkÐkutpvpwiKmokÐtPm<mfmfcoedmolk s.]es>atcumoh`pvimfilcupvPìilP¼cuilPyt ÓedPm^`P¼ilPjkoilpCq`s>i[ìCknmÌedq`si[jiCg~t>[`k4pve É >pr^`p

I[ì

(G,w) Ë P©yeÑko>pC^`s>iKc<sà]_eds>atdcomoh`pwi[ì¥cfePzko^jcG

ilPÊmoilpwg`ks>Pèdcoi%gèdc<cfedgjgtcnmÑs>ermÌeds>s>i[ìG t#)sà]ilPmocoli

Gilkum4^`P)acoedg`hì Kt¶Ìh`tdco[eds !Xt^`c ltdpvpwilPKilc sà]_eds>adtcomfh`pwi edsl^`si1^`P)tco[`cui

[U]ls>pw>Pemo>tPÑgecZedmv1, . . . vn

[ìG

ilPzmfiKpvgjkÐs>Pèdcoi ydt>c1¨ X¬ tdm w(vi)siIg~t>[`k,[`^

kotdpvpwiKmvi g~t^`c i ∈ 1, . . . , n

"!$#%

µiIg~t>[`kNpve É >pr^`p [U]_^jPì Ks>>^ì4[ìG

iKkumPltP`knmfcu^`mx[ìIs>enZe0KtPku^`yedPmoi tm W ′ = 0im

s(vi) = 0gtd^`cxmot^jm

i ∈ 1, . . . , n !Xt^`cNmft^jm i [ji 1n ku w(vi)−W ′ + s(vi) > 0 edstcokNtPltds>tcu>i

vi

ilP)cotd^ài iKm4eibmfiW = w(vi) + s(vi) iKm4tP)eutd^jmfi

w(vi)−W′ + s(vi)

s(vj)g~t^`cÐmftd^jm

vj ∈ N(vj) ∩ vi+1, . . . , vn & êdP`[mft^jksilkkotdpvpwiKmok1tPmmf\mfcoedmolk kotm K = ∅ img~t^`c Ìhèd^ìi[ì

n

1 ku viilknmcot^jai1iKm4ed[oeXliKPm4Cmot^`k<silk<kotdpvpwiKmokÐ[ìK edstcok K = K ∪ vi ^ Peds sà]_edsatcomfh`pwi

coiKPytis>e;ls>^ì1[ìgtd>[`k<pve É >pr^jpK

[ì(G,w) Ë PytdmX>pwpvK[`±emfilpwilPmµ^ìxsà]_eds>adtcomfh`pwiºkK]_i É l^jmoi<ilPmfilpwg`kXs>Pèdcoi e ltcucoibmftPy|>iKPm

[`^ Zedm%|^jiwstcoku^ìÑ|^jiws.]_tdPmocfedmfirsi\kotpwpwiKmvi s±ez[jlcoiKPli

W ′ − s(vi)ilknmgjcobK>kulpwilPm

s±e)yeds>iK^`c^ìsà]_eds>atdcomoh`pwirtcoa>PèdsÒ[ìb¥cfePedg`g`s>^`G

ed^`coedmkot^jkumfcoedmCc w(vi)

[`^ltdpvpwilPKilpwilPm,u^`ku|^]ed^mocfedmfilpwilPm1[ji

vi ? >Pjko P`tdmocoiÑedsatcumfhjpvikopr^`s>isà]_eds>adtcomfh`pwi[ì¥cfedP ko^jc

GiKm4ilknm Ktcocuibm

»ÏR¾¡¿|QSYKJFL L ] ZW_Q i ' ` ]$31)- ( + - Z ] T S2[SNW ]CW34 f Q T ] U S] ` Z f R U 4$YXR+W ] f[T W ] f3 WY[Z \ ] f aV] Y 2& Y f_f ] + 4 ` Q Uhf SWYXSNR Q U "!@t^jc ltdpvpwilPKilc tdPqes l^js>iilPmoilpwg`k4s>>Pjedcoi%mftd^`kIsilk4kotpwpvimfk4[ì[ìlacu2iKmmot^`ks>iKk

C5

[ìG

iKPÊ^jmo>s>kfePm,sà]_eds>adtcomfh`pwi[j^©moh`ltclpwi «· G

ilknm/lt¶Ìh`tcu[eds eds>tcukÕtPeg`g`s>^ìsà]_eds>atdcomoh`pwiÐ[`^silpwpvi « g~t^`cxe ydt>cÕsi4culku^`smfemilP¼moilpwg`k%s>P`led>cui G ilknm1^`P

C5 es>tcuksirgtd>[`kpe É pr^`p [U]^`Pì;Ks>>^ìgiK^jm mfcoi;leds K^`s>ilP)mfiKpvgjkpKtP`knmÌedPm >PjtP kutm u ^`PÑkutpvpwiKmâ[ì<[ìladco

2[U]^`P

C5 sjÓre[jil^ É ls>^ìlkÒpve É >pveds>iKk@gèdcÒ>PKs>^`ku>tP[ìG

^`XltPmfilP`PjilPmu ltcucoilkugtdP`[edPm@ed^ É [jil^ É ec$Kmoilk>Phl>[jilPmfiKk

u µiÒg~t>[jkµpe É pr^`p [ìmftd^jmfiKkPliKkPKs>>^ìKkxg~il^¸m/Kmocoi edsl^`s4iKPmfiKpvgjk

O(m) t^jmoilkxs>ilk,ed^jmocoilk ls>^ìKk,pve É >pveds>iKkâ[ìG

kotPm1[ìKk ls>^ìlkI[`âcfedgjhìClt¶Ìh`tcu[eds@tqjmfiKP^ìlkedg`clk1eydt>c%ku^`g`g`cu>pwmot^`k1s>iKkIkutpvpwiKmok[ì)[ìKaco

2[ìKk

C5

[ìG µil^`c\g~t[`k\pe É pr^`p giK^jm Kmocoi'es l^js>ziKP'moilpwg`k\s>Pèdcoie ydib

sà]_edsatcomfh`pwi4[`^Êsilpwpvi « jR¾¡R~WZWZ£`Z¡Y JFL FL F] 1 R f S] TeU Y QW8R+S\ ` ] R U 4$YXR+W ]dZ Q T W3 ]dZW Q i ' ` ] $%1 - () + - f[T W ] f3 WY[Z \ ] fcf Y Uhf P5

][S ] ` R U a T R+S /10 0 ) ;>FÐ7ÐH ¾M ¯=#; àE@J ZHK= L7%:<H# "

Ë P¼yil^¸mcolkut^`[`cui ()$3()+#() () + -ku^`c1s>iKkIacoedg`hìKk edcoeXKmolcu>kumo>^ìKk[ìKkP`t|il^j[`kg`coiKpvilcok ()%$ )%$ )+* x >59 9b@ *'& .?1 @:9;| 0 ~3 1h . 6359;| 02G?1?>A4:9

Í iKmumfinls±ekokoiiKkum4^`Pì ls>edkokui[ìacfedgjhìlkgedc"Zedmok ledc lt¶Dq`>gèdcumo>k ÕY¸QSQSYrJFL 7'FL O 0+ OFQR+S

G TVU WY[Z \ ] ZhYXW YXR+SP][S w TVU ] Q U 2[SR Q U aV] Z QRa fpf[T W ] f YXW 2[S] faV]G2l QW f χw(G) = ωw(G)

][Sκw(G) = αw(G)

µil^`c%Pjtprq`cuiCÌh`cotdpemo>^ìgtPj[`lcurilknm1[`tPClaeds@ed^g~t>[jkpve É >pr^jp [U]_^jPì;ls>^ì Ë P

eb[`5ozy|^¦ilP¦koibmftP « Û]>sâg~t^jyedm Kmocoi edsl^`srilPmfiKpvgjk1s>>Pjedcoirku^`c1s>iKkadcfedg`hjilk lt¶q`gedcumo>k/Kt^`g`slk

37% 5 : #" % A e P5+ C !$#

()%$ )%$ )w( x >59 9b@q( & .?1 @:9C9 14V|e3 G@:9!ºedcoed[`t É eds>iKpviKPm g~t^`c limomfiPls±ekokoi,[ìÐacoedg`hìKkâs>iNcoKko^`smÌemºPU]_iKkumÕgekâ>pwpvK[`±em Ë PrpwtPmfcui

^ìCsigjcotq`slpwig~il^¸mcKmocoicolkuts>^ÊilPmoilpwg`kpltP`knmÌedPm1g~t^`cpÌhed^ìCacfedgjhì[jimÌe>s>siO(1) kotP©edkuko^`pwi|^jisilk<tg~lcoemftP`kÐku^`cs>iKk<P`tprq`cuilkilPmo>ilcuk<koi ¤tPmilP)mfiKpvgjk

O(1) !@t^`c Kils>e tPkoicoedp KPì\^`PÊgjcotq`slpwi%[ìg`cutacfepvpvemftPzs>P`led>cuiIilP©P`tpCq`coiKk4iKPmf>iKcok Ë P É i^`P©acfeg`hìG = (V,E) iKm w : V → N

^`Pì ¤tPmf>tdP©[jig~t[`k tmI

sà]ilPjkoilpCq`s>i\[ìKkknmÌedqjs>ilk1pve É >pved^ É [ìG Ë P ltP`knmfcu^`msi\g`cotqjs lpwir[ìÑg`cutacoedpÑ¶

pvemftPbs>PèdcoiilP©P`tprq`cuilkilPmo>ilcukko^`ydePm pvP`>pw>kuilc ∑

I∈I xI mfiKsµ^ì∑

I∈I:v∈I xI ≥ w(v)g~t^`cÐmot^jm

v ∈ V xI ∈ N

g~t^`cÐmot^jmI ∈ I.

Ë PÊ[`KkoaPìgedc −→x s>i%ydibKmoil^`c ltPmoilPedPm4g~t^`c Ìhed^ìI ∈ I

s±eryeds>iK^`cxI tm

zs>eyedsil^`cNtgjmo>pveds>i<[ì liIgjcotq`slpwi[jiIg`cutacfepvpvemftPÑs>Pèdcoi<iKPmo Kcoi e stcok

ziKkum

laedsâed^Pjtprq`cuir[ì;ltd^`s>iK^`cok%PjbliKkokoed>cuilkgtd^`ccltds>tcu>ilc(G,w) âtPev^`Pì;ltds>tcoemftP

C ′[ì

(G,w) eds>tcuk KtP`kumoco^`ms±e Kts>tdcfemo>tPC

ilPcoiKpvgjs±e0ledPm Ìhed^ìÑkumfedq`siI ′

[ìC

gedc%^`PknmÌedq`sipve É >pr^`p

IKtPmfilPèdPm

I ′ C iKkumlaedsilpwilPm^`Pìnltds>tcoemftPb[ì(G,w) im%e\s>ipjlpwi1P`tprqjcoi

[ìplt^js>il^jcok tdm xI

s>iP`tdprq`coi[ìpltd^`s>iK^`cokxedkoku>aPjlilk,I 7 s>tcuk

xI

ilkum,^`PiKPmf>iKcÕP`tPPjlaemo !@td^`c Ìhèd^ìv ∈ V ltdpvpwi

veve^pwt>P`k

w(v)Kt^`s>iK^`cok ∑I∈I:v∈I xI ≥ w(v) ∑I∈I xI

ilknmlaedsèd^ÑP`tdprq`coi,[ì lt^js>il^jcokÒ[ì

C PyiKcokoiKpviKPm kotm −→x ^`Pì<kotds>^jmo>tPrtdgjmfpedsix[j^wgjcotacoedpwpvis>Pè>coiÕiKPrP`tprqjcoilk@ilPmfilcuk !Xt^`c Ìhed^ì I ∈ I tPCgiK^jmâg`cuilP`[jcoiÐ^`PCilP`kuilprq`si,[ì

xI

Kt^`sil^`cukg~t^`ctKts>tdco>iKcsilkrkutpwpvimfkC[ji

I âÍ tpwpwiI

ilknm\^`P knmÌedq`si iKm\^ìzg~t^`csÌhed^ì)kotdpvpwiKmv ∑

I∈I:v∈I xI ≥ w(v) liKs±e\P`t^`k[`tdP`Pì^`Pìnltstcfemf>tdPz[ì(G,w) eydib z Kt^`s>iK^`cok eÊcoiKs±e É emo>tP[`^Sg`cotqjs lpwi eg`c$KkCko^jg`g`coiKkoku>tP[jis±e'ltdP`[`mo>tP j ilknmrsiwgjcotq`slpwiÑ[ì

g`cutacfepvpvemftPs>P`led>cuiIku^`yedPm

pw>Pj>pw>koiKc ∑I∈I xI « mfiKsµ^ì

∑I∈I:v∈I xI ≥ w(v)

g~t^`cÐmot^jmv ∈ V

tm −→x′

^`Pjikuts>^¸mf>tdPbtgjmfpes>iI[jinKiKmumficoils>e É emftP iKm4kotdmz′ =

∑i∈I x

′I

tdmotP`k@|^ji,g`^`ko^ìxs>iÕacfeg`hìG

ilkumA É lixg`cotqjs lpwiÕ[ì,gjcotacoedpwpemo>tP1s>P`led>cuiÒgtdkok$K[ì^`P P`tdprq`coi ltPjkumÌePmÑ[jiyedco>edq`s>iKk s>ivP`tprq`cuiz[ìzknmÌedqjs>ilkCpe É ped^ É [ì

G iKm\^jP P`tprq`cuiltdP`kumfedPm[ì;ltPmocfedPmoilk s>irP`tdprq`coi\[ji\kotpwpvimfk%[ji

G sºg~il^¸m[`tP;mfcoi\culkuts>îlPmoilpwg`kltdP`kumfedPm gedc<i É iKpvgjs>iilP©lP^`pwlcoedPmim4ilkokoeÓedPm4mft^`ksilk<kotdpvpwiKmok<[`^Êg~tsÓ l[jcoi[` `P`mgecs>ivg`cotdacfedpwpwi Ë P g~il^jm\ lcu>coivs>ivg`cotadcfedpwpviws>Pè>coiwkotd^`krs±er¤tcopwi

maxcx : Ax ≤ b mfiKsâ^ì Ìhed^ìwlslpwilPm1[ìÑs±evpemocoliA

iKkumkotm0 kotm 1 tm ∆

s±evyedsil^`cIpe É pr^`p [ìlk

"!$#%

[`mfilcupvPedPmfk\[jilkwkot^`k5¶3pvemfcu Kilk\[ì)s±e¼pvemoco KiA ∆

ilknmvq~tcuP`li)gedcÑ^`PjirKtP`knmÌedPmfi tmk = |I| Ë Pg~il^jm,pedPmfilPèdPmx^¸mf>s>kuilcÒ^`Pvcolku^`smfemx[ì Í t|t °iKcfedcu[`k ÌhjcoÔnydilcxiKm edcu[`tk simoh`lt¶c$Kpvi ³ \[edP`k¨ Ìhª²¬ Í imoh`ltdc$lpwiP`t^`k4[`m|^jiCs>i1gjcotq`slpwi[ìg`cotadcfedpwpemf>tdPzs>PèdcoiilP©P`tpCq`coiKk4iKPmf>iKcok4e\^`Pìkots^jmftPbtgjmo>pveds>i −→

x′′ mfiKs>si%^ìg~t^`cÐmot^jmI ∈ I |x′I − x′′I | ≤ q∆

!Òcoilpw KcoiKpviKPm tP¼mocot^¸yi\^`PìCkots^jmftP¼tgjmo>pveds>i −→x′

[`^g`cutacoedpvpwis>>Pjedcoicoils>e É È P|¶ko^jmfitP¼KP|^jp KcoiCmot^`k%silkydibmfil^jcok1[U]ilPmfilcuk −→x′′

mfiKs>kI^ìCgt^jcmftd^jmI ∈ I |x′I − x′′I | ≤ q∆ !ºecopwUs>iKk,yi KmfiK^`cok^`yKco ilPm4s>iKk ltP`[jmftP`k tP'Ìh`t>kum<siIyi Kmoil^`c4kots^jmftPzpw>P`pr^`p !ºecvsi©moh`ltclpwi ³ [ìS¨ Ìh`ª²¬ ]_iKkumb^jPìÊkuts>^¸mf>tdP tg¸mf>pvedsiz[`Âgjcotacoedpwpvi)s>Pè>coiilP

P`tpCq`coiKkNiKPmf>iKcok |Í iKmomoipwKmoh`t|[ì<g`coiKP`[^jPwmoilpwg`kdltP`knmÌedPm edc q im∆

kotPm É lkxkos>i<adcfedg`hjiilknm É [`tPsi1P`tprq`cui1[ì1yi KmfiK^`cokIrmfilknmfiKciKkumpltPjkumÌePm OPì,>pwg`s>KpviKPmfemftP%mfiKs>six^ìKs>s>iN[ì/KiKmumfi,gjcotVll[j^`coiÐPjiÐkoiKcfeIgekºg`coemf|^ji g`^`>kuÛ]_s|Óe4[ìsà]tcu[`coix[ì

(q∆)qyibmfiK^`cokÕmoilknmfilc È Prg`coemf|^ji tdPrgiK^jmâgèdcºi É iKpvgjs>iÐculkotd^`[`coi,s>i,gjcotacoedpwpvi

s>Pè>coi g`^`kxkui%koiKcuy|>cx[ìIs>ekuts>^jmo>tPjltdpvpwig~tPmx[ì[`lgecumÐ[]_^`Pji%g`cotVlK[`^`cui qjcfedPÌh)eP`[q~t^`P`[

Í tpwpvi>s¸Ówe^`PvP`tprqjcoi ltPjkumÌePm,[ìacfeg`hìlkN[ìmÌeds>siÐqtdcoP`Ki tPve[`tdP^jPzedsatcomfh`pwiilPbmoilpwg`k

O(1)g~t^`cÐmocot^¸yilcsi%P`tprqjcoi Ìh`cutpemf>^ì1g~tP`[jlco1[]_^`P©acoedg`hì1[ì%mfed>ss>i4qtdcoP`Ki

Í ]ilkumIed^`kukoµsi edk[ìKkacoedg`hìKkkog~bl ^ìlk |^jXtdPm4e^Êg`s^`k9kotdpvpwiKmok Ë Pez^jPeds>atdcomoh`pwiiKP¼mfilpwg`k

O(1)g~t^`ccleds K^`s>iKcIs>iCP`tprq`cuiCÌh`cutpemf>^ìCgtPj[`lcu\[ìlk

acoedg`hìKk<[ì%mÌeds>siq~tcoPj >P`ku tP)giK^jm<coKkot^j[`coi()$3()+#() ¯ilPbmoilpwg`k<s>Pèdcoiko^jcÐs>iKk,acoedg`hìKkÐgtd^`cxsilku|^jils>k,s>ilkxacfeg`hìlk,g`coiKpvilcok,edg`gecfedkokfePmx[edPjkÐs±e[`bKtpvg~tkumftPwpvt¶[`^`s>ed>cuiItPm^`Pjimfed>ss>i4qtdcoP`Ki ¾Í i ledpwls>tcuiI^`Pì1cuilpvedco^ì1[ji >edcopw>[bim'4iKil[¦¨ ·d·¬ ^`¾tq`kuilcuyedilPmÐ^ì4s>ig`cutq`slpwi ($%()+#( () + - g~t^jyedmmfcuiculkotds>^wilPmfiKpvgjk4g~tsÓ¸Pjtpv>eds~ko^`cs>iKk<acfeg`hìlk[ì1mfed>ss>i KtP`knmÌedPmfi ()%$ )%$ ) x >59 9b@

tmG

^`P acoedg`hìz[jizs±eqKs±edkukoi ¸ eyi b^`PìD¤tPhKmftPS[jibgtPj[`lcoemftPw !ÒcoiKpvlcuilpwilPm

tPedsl^`siiKPmfiKpvgjks>P`led>cui%mft^`ks>iKkkutpvpwiKmok[ìC[ìKaco2iKmImot^`k4s>iKk

C5

[ìG ilP¼^jmo>s>kfePm

sà]_edsatcomfh`pwi4[`^©moh`ltdc$lpwi «· G

ilknm<kfedPjkC5

P`[`^`m G iKkum<[`tdP1^`Pbacoedg`hìcKt¶Ìhjtco[es im χw(G) = ωw(G)g~il^¸m Kmocoi

es l^js>1ilP)mfiKpvgjk<s>>Pjedcoi%gec<sà]_eds>atdcomoh`pwig`culkuilPmfed^©s>ilpwpwi « j >PjtP G ∈ Gk eyi k ≥ 1 G ilkum^jPC5 eds>tcukIs>emfibÌhjP`>^ì\^jmo>s>koKiC[eP`k1s>isedk1[ìKk

acoedg`hìKkkug l ^ìlk gjcolkuilPmfKiv[èdP`ks±ezkuibKmo>tPg`cubKl[ìKPmoi g~il^jmnKmocoi\^jmo>s>koKirg~t^`c es l^js>ilcs±eCKts>tdcfemo>tPzg~tP`[jlcoKipw>Pj>pr^`pGiKP©moilpwg`k ltP`knmÌedPm ¥º>Pèds>iKpviKPm kutm C = (v1, v2, v3, v4, v5)

^`PC5

[jiG imvkutm

v1

imv3

koilkwkotdpvpwiKmok\[ì[ìKaco

2 !@edc,[j P`mftP[ìKkÐacoedg`hìKkÐ[ì%s>enKs±edkukoi j sà]ilPjkoilpCq`s>iA

[ìKkÐkotpwpwiKmfkxP`tPbe[oeXliKPmfkC

iKkum^`PiKP`koiKprq`sirkumfedq`s>i B = V \ (C ∪ A) = N(v2) \ C = N(v4) \ C = N(v5) \ C imG[B]

ilknm^jP Ktacfeg`hì È P ltdP`koK|^jilPKi>s@Óebg`co lkolpwilPm4ilPjkoilpCq`s>iKkknmÌedqjs>ilkpve É >pved^ É[ì

GltdPmoilPePmzed^'pvtP`k\^`P [ìlkÑmfcut>kÑkotdpvpwiKmok

v2 v4

t^v5

NsÐkl]_edamw[ìv1, v4 ∪ A

ÜG

ÚuÝZßáã ( HDí ì25ÚºÝ.ì -/5Ü /êláÚ 7∆ ≤ 3

î

37% 5 : #" % A e P5+ C !$#

v2, v4 ∪ A v2, v5 ∪ Aimv3, v5 ∪ A t^`kzsilkbed^¸mfcoiKkzilPjkoilpCq`s>iKkzkumfedq`s>iKkzpve É >pved^ É>Phls>^jilPmsà]ilP`kuilprq`siÑkumÌeq`s>i

v1, v3[ì

C !Òs>^jkgjcobK>kulpwilPm v1, v3 ∪ A′ ilkumC^`P¦ilP`kuilprq`sikumfedq`si[ì

GkoXiKm%kuil^`silpwilPm%ku

A′ ilknm%^`P¼iKP`koiKprq`siCkumfedq`sipve É >pvedsµ[ìG− C tdmftPjkI^ì

v1, v3 ∪ Ailkum4^`P©ilPjkoilpCq`s>ikumfedq`si1[ì

G 5ÕY¸QSQSYmJFL )?LpOFQR+S

k ≥ 1^ ][S

G ∈ Gk

OFQR+SC = (v1, v2, v3, v4, v5) TeU C5

aV]G

S]& k T ] v1

][Sv3 f QR ] U S aV] aV] W4 2

,OFQR+Sw TeU ] Q U 2[SR Q U aV]Z QRa f f[T W ] f f Q `;` ][S f aV] G l QW f R?] 1 R f S] TeU ]2_Q QWYXSNR Q U Z Q U a,4[W4_] ` R U R `;TV` S aV]

(G,w)YX]32 ZW3432[R f 4 ` ] U S max

(w(v2), w(v4) + w(v5)

)] Uhf ] `ji ] f f S Y i ] f 2_Q U S] U Y U S/Y Tr` QR Uhf TVUqf Q `;` ][SaV]

v2, v4, v5

4 ` Q Uhf SWYXSNR Q U tm S ^`Pì;Kts>tdcfemo>tPÊgtPj[`lculi\pw>P`pr^`p [ì(G,w) tm C = (v1, v2, v3

v4 v5)^jP

C5

[ìG

moilsN|^ji ku G 6= C5 v1

iKmv3

kut>iKPmCs>iKkCkotdpvpwiKmok[ì[ìlacu2 !Ò^`ko^ì

N(v1) \ C = N(v3) \ C = ∅imN(v2) \ C = N(v4) \ C = N(v5) \ C = B tP¼ko^jg`gtdkoir^ì

mftd^`ks>ilk<iKP`koiKprq`silk4kumÌeq`s>iKk4[jiSKtPmfilP`PìKPmkutm4e^l^`P kotm[ìl^ É kutpwpvimfk<[ì

C È P KtP`kul^ìlPhliÑtPKmo^`[ìCs>iKkcltds>tcoemftP`k4gtPj[`lculilk1[j^C5 C = (v1, v2, v3, v4, v5)

[ìGeyi ;ltpwpvig~t>[`k4g~t^`cIsilkkotpwpvimfk

w t#mftd^`kIsilkIiKP`koiKprq`silk%knmÌedq`silk%kutPm%[ìKk1P`tP¸¶3edc$mfiKk[ì

C Ë PAsilkwedg`g~ilss>iKcfe[ìlk ltstcfemf>tdP gtPj[`lculilkvgedcnmf>iKs>silk\[ìC tdmftdP`kv^ì©mft^¸mfilkws>iKk

ltds>tcoemftP`kÕgtdP`[`lculiKk<[ìC = (v1, v2, v3, v4, v5)

[`tydilPm<g~tkokul[ìKced^zpwt>Pjkw(v2)

iKP`koiKprq`s>iKkkumfedq`silk ltPmoilPedPm

v2 iKm4ed^©pvtd>P`kw(v4) + w(v5)

ilP`kuilprqjs>ilkknmÌedq`silk ltdPmoilPePmv4

t^v5 tm

S ′ ^`Pì'ltds>tcoemftPg~tP`[`Kcoli©[ìG′ KtPmfiKPedPmsibP`tprqjcoibpw>P`pr^`p gtkukoq`s>ib[U]ilP|¶

koiKprq`silk%knmÌedq`silkS

eyi S ∩ v2, v4, v5 6= ∅ tdmftdP`k

qsiCP`tdprq`coiC[U]_iKP`koiKprq`silk1kumfedq`silk%[ì

S[ìS ′ eydib S ∩ v2, v4, v5 6= ∅ iKm%ku^`g`g~tkotdP`k%^ìCKtPmfcfe>coiKpviKPm%wsà]_e *wcopvemftP)[`^¼s>iKpvpwi

q > max(w(v2), w(v4) + w(v5)

) tm c(v) s>i<Pjtprq`cui<[U]_iKP`koiKprq`silkÕkumfedq`s>iKkÒ[ìS ′ ltdPmoilPePmxs>ikotdpvpwiKm

v s>tcuk q > w(v4)+w(v5)pvg`s>^ì

c(v4) > w(v4)t^

c(v5) > w(v5) eP`kÒgiKcumoi<[ìaKP`lcoeds>mf tPku^`g`g~tkoi

c(v4) > w(v4) È PqltdP`koK|^jilPKir>sµi É kumoi^`PÊilP`kuilprqjs>ikumÌeq`s>iS ′ ∈ S ′ltdPmoilPePm

v4 imtP¦ezkotmS ⊆ v2, v4 ∪ A kotm S ⊆ v2, v3 ∪ A eP`k1s>iKk1[ìl^ É edk tPcoiKpvgjs±eXKi

S ′ gèdc v1, v3 ∪ A Í ils±ew[`>pw>P^ì[ì1siP`tdprq`coi[U]ilP`kuilprq`silk%knmÌedqjs>ilk ltPmoilPedPm

v4 iKmÕyiKPmo^ìlss>iKpviKPmN[ì1s>iÐP`tdprq`coi<[]_iKP`koiKprq`s>iKkÕkumÌeq`s>iKkltPmoilPedPm

v2 P`ko`Pjt^`kÒtqjmoilP`tdP`k^`Pì Kts>tdcfemo>tPgtPj[`lculi

S ′′ [ì G eyi q−1

ilPjkoilpCq`s>i<knmÌedq`silkS

yKco ¾edPmS∩v2, v4, v5 6= ∅ Í ibKltPmfcuil[`ms>i Ìh`td É [ì

S ′ !ºedc KtP`koK^ìlPm q = max(w(v2), w(v4) + w(v5)

) R¾¡R~WZWZ£`Z¡Y JFL FLjOFQR+S

G TeU C5 (v1, v2, v3, v4, v5)^q][S

w TVU ] Q U 2[SNR Q U aV]rZ QRa f f[T W ] ff Q `;` ][S f l QW f Rn] 1 R f S] TeU ]$2_Q QW$YXSR Q U Z Q U a,4[W4_] ` R U R `;Te` aV]

(G,w)YX]32jZW3432[R f 4 ` ] U S

max(w(v2), w(v4) + w(v5)

) ] Uhf ] `ji ] f f SY i ] f 2_Q U S] U Y U S?Y Ts` QR Uhf TeUtf Q `;` ][S,aV]v2, v4, v5

!@t^jcmfilPj[`coiw^`Pìjltstcfemf>tdP¼gtdP`[`lculiwgedcnmf>iKs>sir[U]^`P

C5

b^`Pìjltstcfemo>tPg~tP`[jlcoKiÑ[ìG koiK^`s>iKpviKPm4[ìK^ É gecfedpjKmocoilkkutPmpvg~tcnmÌedPmfkI

Ç si4Pjtprq`cui4[ji ltdg`>iKkN[jiv1, v3

[eP`kxs>e ltstcfemo>tP\g~tP`[jlcoKiIgecumfilss>i [`lPjtdmfgedcs imÇ siP`tdprq`coi[U]ilP`kuilprqjs>ilkkumfedq`s>iKkI[`^

C5

[`KcoilPmokI[ìv1, v3

[ìCs>ejKts>tdcfemo>tP)gedcnmfils>si [`KP`tdmogèdct !@td^`c ÌheXl^jP [ìKks

ilPjkoilpCq`s>iKkwknmÌedqjs>ilkv1, v3

[`^C5 Ìhed^ì v1, v3 ∪ A′ t# A′ ilknmw^`P

ilPjkoilpCq`s>ikumfedq`si<pe É peds|[ìG−C iKkumN^`PwilP`kuilprqjs>ipe É pes¸[ì

G|^jKmfiKP`[

v1, v3 !Xt^`cÌheXK^`P¦[ìlktilPjkoilpCq`s>iKk1kumÌeq`s>iKk

S[`KcoiKPmok1[ì

v1, v3[`^

C5 S ∪ A iKkums>iCkoil^jsâilP`kuilprqjs>ikumfedq`si1pe É peds[ì

GmfilPj[edPm

S

"!$#%

!ºedcbsisilpwpvi «³ g~t^`cDÌhèd^ìC5

[ìG

s4Óe^`Pì=Kts>tdcfemo>tP gtPj[`lculipw>P`pr^`p[ì

Geydib

t = max(w(v2), w(v4) + w(v5)

) iKP`koiKprq`silkrknmÌedq`silksltPmfiKPedPmÑed^SpvtP`k^`P[ìKkkotdpvpwiKmok,[ì

v2, v3, v4 È P)g`cuilPePmÐ^`Pì1[ìcKilk Kts>tcoemftPg~tP`[jlcoKi%pw>P`pr^`p tPbgiK^jm<iKP¸¶s>iyiKc©silkbkutpvpwiKmok

v1iKm

v3[jilk)ilP`kuilprqjs>ilk©kumfedq`silkEltdPmoilPePmÊs>iKkb[ìl^ É kotpwpvimfk u^`koÛ]_tqjmoilP`c¼s±e g`s>^`kgimfmoiSyeds>iK^`cg~tkuko>qjs>ig~t^`c

s[edPjk^`Pì ltds>tcoemftP gecumfilss>i¦[ì

Ce ydib

t = max(w(v2), w(v4) + w(v5)

) !ºedc\siDltdcotss±edcoi «ª >sxi É >knmfi)^`PìEKts>tdcfemo>tPg~tP`[jlcoKigedcnmfils>si¼eyi

χ(G[C])iKP`koiKprq`silk©kumÌeq`s>iKk mfiKs%^ì

max(w(v2), w(v4) + w(v5)

) iKP`koiKprq`s>iKkkumfedq`silk/KtPmf>iKP`PìKPmed^bpwtP`k,^`Pbkutpwpvim<[ì

v2, v3, v4 imed>Pjkos = χ(G[C])− t

g~il^jm mfcuies l^js>li1iKP©moilpwg`k ltPjkumÌePm Ë PÊg`culkuilPmfipedPmfilPèdPms.]es>atcumoh`pvi4^`2es l^js>i^`Pì ltds>tcoemftPbgtdP`[`lculipvP`>pC^`pG[ì(G,w)

g~t^`cÐ^`P)acoedg`hìG

[ìGk k ≥ 1 sUiKP`s yied^bg`s^`k

k¤t>kx^`P

C5g`cubleds K^`s>1e^©adcfedg`hji

ltd^`cfedPm u^`ku|^]DKir^ìrs>iacfeg`hìrcuilkumfedPm1PU]_edmg`s^`kI[ìC5 im1[`tPkotmcKt¶Ìh`tdco[eds , s>tcuk ^`Pì ltds>tcoemftPpvP`>pr^jp g~tP`[`KcoKi,[`âcfedgjhìdlt¶Ìh`tco[èdsgiK^jm mfcuixtq¸mfilP^ìxilPmfilpwg`kXs>Pèdcoi

ilP©^jmo>s>kfedPm<s.]es>atcumoh`pvig~t^`cs±eCKs>|^ji1gtPj[`lculipe É pr^`p g`coKkoilPmoCed^©s>iKpvpwi « ¸ Ìhed^ìIKmÌegi d]ilknmC[`coi|^èdP`[btPzku^`g`g`cu>pwiI^`P

C5 C = (v1, v2, v3, v4, v5)[`^badcfedg`hji

ltd^`cfedPmG′ eydibs±eC¤tPmf>tdP©[jig~tP`[jlcfemf>tdP

w′ s.]edsatcumfhjpvi4g`cotVbK[ì[ìs±etZe0KtPÊku^`yedPmoiw>s leds K^`s>i ilPSmoilpwg`knltP`knmÌedPm ^`Pji ltds>tcoemftPgtdP`[`lculivpvP`>pr^jp gedcnmfils>si\[ì

Cmoilss>iÑ^ì

t = max(w(v2), w(v4) + w(v5)

) iKmNmfiKs>si<|^jiskutmNsi4gjs>^`kÕg~iKmfmN|^jig~tkoku>q`si 7 s>tcukÒ>silPjs Kydi

mftd^`kxsilkNkotpwpwiKmfkN[ìC iKmx>siKP`s yi4mft^`kxs>iKkxkutpvpwiKmokÕ[ì

Aeyi I^`Pzg~t[`kÕ>Ph¤Kco>iK^`cÕt^Kaeds~

t iKm<kut^`kumocfedmte^ É g~t[`kx[jiImft^jk,s>iKkÐed^¸mfcoiKk,kutpvpwiKmok,[ì

A Ë PzP`tdmoiG′′ s>iP`t^jydied^badcfedg`hji

tqjmoilP^ iKm w′′ s±eP`t^jydils>sip¤tdPKmo>tPv[ìIg~tP`[`Kcfemo>tP V s>tcukxs.]es>atcumoh`pvixcoKkot^jmÐcobK^`cokuydilpwilPms>igjcotq`slpwimltstcfemf>tdPpvP`>pC^`p ku^`c

(G′′, w′′) È P P siP`tprq`cuipvP`>pC^`p []_iKP`koiKprq`s>iKkkumfedq`silk[edP`k^`PjinKts>tdcfemo>tPz[ji

(G′, w′)ilknmItq¸mfilP^©gedc<s>e;¤tdcopr^`si%ko^jyedPmfi

χw′(G′) = t + max(s, χw′′(G′′)

).

Ò]edsatcumfhjpviÕilPjs Kydi<ed^rgjs>^`kk ≤ n

¤t>kX^`PC5 dÍ hed^ì Kts>tcoemftPgtdP`[`lculi<gecumfilss>iÒ[U]^`P

C5

giK^jm Kmfcui leds K^`s>Ki<ilPÑmfiKpvg`kdKtP`kumfedPm !@td^`cÕs>iacfeg`hì lt¶Ìh`tcu[edsPèds s>ecltstcfemo>tP\g~tP¸¶[`Kcolipw>P`pr^`pGg~il^jmpKmocoi leds K^`s>Ki1ilP)mfilpwg`k<s>P`led>cui V >Pjko s>i%moilpwg`kÐmftdmfedsFKiKm4sà]_eds>adtcomfh`pwig~t^`c<s±e;Ks±edkukoi ilkum4s>P`led>cui È P©coKko^`pw tP)tq¸mf>iKPmR¾¡R~WZWZ£`Z¡Y JFL 62L ] 1 R f S] TVU Y QW8R+S+\ ` ] R U 4$YXR+W ] Z Q T W ]ZW Q i ' ` ] ()$3()+#() ( + f[T W ] f W$Y[Z \ ] f f Y Uhf P5

][S ] ` R U a T R+S*

/10 0 ) ;>FÐ7ÐH ¾M ¯=#; àE@J ZHK= L7%:<H "

Ë PyiK^jm%culkotd^`[`coi +%() - $%1)- 4 () + -ku^`cIsilk4acfeg`hìlk edcoeXKmolcu>kumo>^ìKk[ìKkP`t|il^j[`kg`coiKpvilcok ()%$ )H)+* x >59 9b@ *'& .?1 @:9;| 0 ~3 1h . 6359;| 02G?1?>A4:9

Í iKmumfinls±ekokoiiKkum4^`Pì ls>edkokui[ìacfedgjhìlkgedc"Zedmok ledc lt¶Dq`>gèdcumo>k

37% 5 : #" % A e P5+ C !$# .

!ºedc4s>isilpwpvi «² κw(G) = αw(G) È P^¸mf>s>koedPm<sà]_eds>adtcomfh`pwiIs>P`led>cui%gtd^`c2- 45- *,$3-4 % *,$3- () +\g~t^`cºs±e Ks±edkukoiÐ[ìKkÒacfeg`hìlk Kt¶3qj>gedcnmfk Kt^`g`s gjcolkuilPmfiKPwkuibmf>tdP « κw(G)

giK^jm4[`tP Kmocoi es l^js>1ilP)mfiKpvgjk<s>>Pjedcoi ()%$ )H)w( x >59 9b@q( & .?1 @:9C9 14V|e3 G@:9

emfi Ìh`P`>^ì,g`coKkoilPmoli<[edPjkâs±ekoi KmftPrg`cubKl[ìKPmoigt^jc leds K^`s>iKc@^`Pji/ltstcfemf>tdPg~tP`[jlcoKipw>P`pr^`p [U]^`P acoedg`hì É bg~il^jmCKmfcuibedg`gjs>>^`Kizko^jc\s>i Ktpwg`s>KpviKPmfed>cuivg~t^`ctedsl^`silcC^jPìgedcnmfmf>tdPzilPrls>^ìlkpw>P`pr^`pGilP)mfiKpvgjkpKtP`knmÌedPm ()%$ )H) x >59 9b@

tmG

^`P acoedg`hìz[jizs±eqKs±edkukoi ¸ eyi b^`PìD¤tPhKmftPS[jibgtPj[`lcoemftPw !ÒcoiKpvlcuilpwilPm

tP leds K^`s>iwilPmoilpwg`kCs>P`led>cui\sà]ilP`kuilprqjs>i[jilkrkutpvpwiKmokC[ìv[ìladco2

iKmrmot^`ks>ilkC5

[ìG ilP^jmo>s>kfedPm<s.]es>atcumoh`pvi[`^)mfhjltclpwi «·

GilkumºkfedPjk

C5

>Pj[`^`m [`tPxilknmâ^`Pradcfedg`hjiKt¶Ìh`tdco[eds κw(G) = αw(G)g~il^¸mKmocoiedsl^`s

ilP)mfiKpvgjk4s>P`led>cuiIgec<sà]_eds>atdcomoh`pwig`culkuilPmfilP©koibmftP « j >PjtP G ∈ Gk eydib k ≥ 1 G iKkum^`PC5 es>tcuk4s>eCmfi Ìh`P`>^ì^jmfs>koli1[èdP`ks>i ek[ìs>e

ltds>tcoemftPÑ[ìlkxacfedgjhìlkxkog~bl ^ìlk g`culkuilPmfKi%[edP`kxs±ekoi KmftP « « giK^jm/mfcuiI^jmo>s>koKi<gt^jces l^js>ilc<^jPìgedcumomo>tPzpw>Pj>pr^`pGiKPKs>>^ìKkilP)mfilpwg`k ltdP`kumfedPm ¥º>Pèds>iKpviKPm kutm C = (v1, v2, v3, v4, v5)

^`PC5

[jiG imvkutm

v1

imv3

koilkwkotdpvpwiKmok\[ì[ìKaco

2 !@edc,[j P`mftP[ìKkÐacoedg`hìKkÐ[ì%s>enKs±edkukoi j sà]ilPjkoilpCq`s>iA

[ìKkÐkotpwpwiKmfkxP`tPbe[oeXliKPmfkC

iKkum^`PiKP`koiKprq`sirkumfedq`s>i B = V \ (C ∪ A) = N(v2) \ C = N(v4) \ C = N(v5) \ C imG[B]

ilknm^jP Ktacfeg`hì È P ltdP`koK|^jilPKi>s@Óebg`co lkolpwilPm4ilPjkoilpCq`s>iKkknmÌedqjs>ilkpve É >pved^ É[ì

GKtPmfiKPedPmÑed^pvtP`k^`P[ìKkmfcut>kkotdpvpwiKmok

v2, v4, v5 kutm v1, v4 ∪ A v2, v4 ∪ A v2, v5∪A

imv3, v5∪A mft^`kÕsilkNed^jmocoilkÕiKP`koiKprq`s>iKkxknmÌedqjs>ilkÕpve É >pved^ É >PKs>^ìKPmÕsà]ilP`kuilprq`si

kumfedq`siv1, v3

[ìC !Òs>^jkg`cublkoKpviKPm v1, v3 ∪ A′ iKkum^`PilPjkoilpCq`s>iÊkumÌeq`s>i©[ì

Gkuim

koiK^`s>iKpviKPmÐkoA′ ilkumÐ^`PbilP`kuilprqjs>iIknmÌedqjs>iIpve É >pveds[ì

G−C F tdmotP`kx|^jiv1, v3∪A

ilknm<^`PilPjkoilpCq`s>ikumfedq`si1[ì

G 5ÕY¸QSQSYmJFLLpOFQR+S

k ≥ 1^ ][S

G ∈ Gk

OFQR+SC = (v1, v2, v3, v4, v5) TeU C5

aV]G

S]& k T ] v1

][Sv3 f QR ] U S aV] aV] W4 2

,OFQR+Sw TeU ] Q U 2[SR Q U aV]Z QRa f f[T W ] f f Q `;` ][S f aV] G l QW f R?] 1 R f S] TeU ]ZhYXW8SNR+SR Q U ] U 2& Rk T ] f Z Q U a,4[W4_] ` R U R `;Te` aV]

(G,w)YX]32 ZW432[R f 4 ` ] U S w(v1) + w(v3)

2& Rk T ] f2_Q U S] U Y U SPY Tr` QR Uhf TVUqf Q `;` ][SaV]v1, v3

4 ` Q Uhf SWYXSNR Q U Í tP`ku>[`KcotP`k<^jPìgedcumomo>tPvilP'ls>^ìlk,pvP`>pr^jpK

[ìG ¾Í hed^ì Ks>>^ì[ì

KKtPmfilPèdPm

v1

t^v3

iKkum4^`P©kot^jkn¶3iKP`koiKprq`si[ìC ^`g`g~tkutP`k,^ì

KltdPmo>iKP`Pì4g`s>^`kN^ì

w(v1)+w(v3)ls>^ìKkKtPmfilPèdPm

v1

t^v3 7 s>tcukÕtP

g~il^jmItqjmfiKP`>c4^`Pìgedcnmfmo>tP)iKP=ls>^ìlkpvP`>pC^`pK

eyi rg`co lkolpwilPmw(v1) + w(v3)

ls>^ìKkltdPmoilPePm

v1

td^v3

ilP edg`gjs>>êPms>iKktdgKcfemo>tP`k[ìzcuilpwg`s±e lilpwilPmko^`yedPmfilk tm c(v) siP`tpCq`coi[ìcls>^ìKkÐ[ì

KltPmoilPedPm

v Ë Pcoilpwg`s>eXlic(v1)−w(v1)

ls>^ìlkK

[ìK

ltPmfiKPedPmv1

gedcK \ v1 im%[ìCp KpvitP¼cuilpwg`s±eXKi

c(v3) − w(v3)ls>^ìKk

K[ìK

KtPmfilPèdPmv3

gecK \ v3

"!$#%

R¾¡R~WZWZ£`Z¡YJFL 2M L OFQR+SG TeU C5 (v1, v2, v3, v4, v5)

^ ][Sw TVU ] Q U 2[SNR Q U aV] Z QRa f f[T W ] f f Q `;g

` ][S f hl QW f R] 1 R f S] TVU ]ZhYXW8SR+SNR Q U ] U 2& Rk T ] f Z Q U a,4[W4_] ` R U R `;TV` aV](G,w)

YX]32 ZW3432[R f 4 ` ] U Sw(v1) + w(v3)

2& Rk T ] f 2_Q U S] U Y U SY Tr` QR Uhf TVU aV] fcf Q `;` ][S f v1, v3

Ë PzKmo^`[`>is>iKkxgecumfmftP`kNilPDKs>>^ìKkxg~tP`[jlcoKi%[U]^`PC5 C

eyi %s±e ¤tPKmo>tPv[ìIg~tP`[jlcfemf>tdPw Ë Pzeg`giKs>si<^`PìIgèdcumomftPÑilP ls>^ìlkNgtdP`[`lculi[`^

C5

^jPìgèdcumomftPÑgedcnmf>iKs>si<ilP Ks>|^jilkN[ìC !@t^`cmfilPj[`coiÑ^`PìÑgecumfmftPgedcnmf>iKs>siCilP ls>^ìKk1[ì

CG kuil^`silpwilPm[ìl^ É gedcfep mfcoiKk

kotdPm4pvg~tcumfedPmfk%Ç sirP`tprqjcoi\[ì;Ks>>^ìKk1[ì\s±ezgèdcumomftP¼gecumfilss>iilP ls>^ìlk1[ji

CltPmoilPedPmkotdm

v1kutm

v3 [`KP`tdmfgecs imÇ si1P`tprq`cui1[ì ls>^ìlkPì KtPmfilPèdPmIgek

v1t^

v3 [jlP`tdmogedct !@t^jcIKmoilP`[jcoir^`Pjirgedcnmfmo>tP)gecumfilss>i1ilPqls>^ìlk[`^

C5 C = (v1, v2, v3, v4, v5)v^`PìCgedcn¶

mfmftPiKPEls>^ìlkÐg~tP`[jlcoKi1pvP`>pC^`p#[ìG tPb[jtm,KmfiKP`[`cui1s>ilk

tKs>|^jilk

v4, v5 v4 v5t^v2

[j^C5

ilPeutd^jmÌedPmI ÌheXK^`Pì\[ìsliKktls>^ìlk^`Pìsls>^ì

B′ [ì G[B] Ë P¼ko^jg`gtdkoi^ì Ìhed^ì4kotpwpwiKm

a ∈ Aedg`gedcoe>mâ[edP`kÕi É eXmfiKpviKPm

w(a)ls>^ìKkÒ[ì

K tPwP`tdmoi<Û]e^l^`Pji[ì liKk Ks>|^jilk ltdPmo>iKPm4^`P©kotdpvpwiKm<[ji

C Ò]edsatcumfhjpvi es l^js>i%^`Pji%gedcnmfmo>tPvilPEKs>>^ìKkÐpw>P`pr^`p#[ì(G,w)

gt^jcÐ^`Pbacoedg`hìG

[ìGk k ≥ 1 iKP ko^`gjg`copedPm

k¤tkC^`P

C5

[`^ acfeg`hìDKt^`cfePm u^`ku|^] libÛ]>sxPU]ÓedmCg`s>^jked^K^`P

C5

[edP`k\sizacfeg`hì iKmwedP`koxilkumw^`P acoedg`hì lt¶Ìh`tco[èds mfedPmÑ[`tP`Pj©^jPC5 C gecs>itltcuts>s>ed>cui d >sXi É kumfi\^jPì\gedcnmfmo>tPgedcumo>iKs>s>iilP Ks>>^ìKk%[ì

Ce ydib

s = w(v1) + w(v3)ls>^ìKkpKtPmfiKPedPmv1

t^v3 im t = κw(G[C])− s

Ks>|^jilkv4, v5 v4 v5

t^v2 Ò]edsatcumfhjpvixcolkut^jmxs>i<g`cutq`slpwi<[ìgèdcumomftP`PìKpviKPmÒilP ls>^ìlkÕg~tP`[`Kcolicubl^jcokoyiKpviKPm tm

G′ sivacoedg`hì lt^`coedPm eyib w′ s>er¤tPmftPS[ìgtPj[`lcoemftP Õ]_edsatcomfh`pwi;es l^js>is =

w′(v1) + w′(v3)iKmt bK^`cokuydilpwilPm s.]es>atcumoh`pvi leds K^`s>i

κw′′(G′ − C) t# w′′(v) = w′(v)g~t^`cÒmft^jkNsilkÕkotdpvpwiKmokv[ìG−C stcokÒsiP`tprq`cuipvP`>pr^jp[jipKs>|^jilkÕ[edP`kN^`PìgecumfmftP

ilPrls>^ìKk4[ji(G′, w′)

ilknmtdqjmfiKP|^©gedcs>et¤tcopr^js>ir

κw′(G′) = s+∑

a∈A

w′(a) + max

t, κw′′(G′ − C)−

∑

a∈A

w′(a)

,

t#A

ilknmÐs.]_iKP`koiKprq`s>i[ìlkxkotpwpwiKmfkNP`tPzed[oeXKilPmfk,C

[edP`kG′ Ë PPjtdmfi|^]_s~Óvei É eXKmoilpwilPm

∑a∈A w

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(G′−C,w′′)^` KtPmfilP`PìKPmx^jPvkutpvpwiKmâ[ì

A iKm,[`tPPìIg~il^¸yilPmgekPKmocoiKmfiKP`[`^ìKkÐilP)e ut^jmÌePmx^`Pì ls>^ì[jiC s~Óe

κw′′(G′−C)−∑a∈A w

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v4, v5 v4 v5t^v2 P`ko s.]es>atcumoh`pvi@ko^`gjg`copvi

k < n¤td>kX^`P

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OPì R~V ´ Y[edPjk^jP¦adcfedg`hji\ilkum^`Pji\gedcumomo>tP¼[jiws.]_iKP`koiKprq`sir[ìlkkutpvpwiKmok%ilP¦[ìK^ É ilP|¶koiKprq`silk

V1

iKmV2

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tPm1s>ipjlpwi4ydt>ku>Pedadi[eP`k

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V1 V2

a

b d

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a, b, c, dimr[U]^`P kutpvpwiKmCed[`[jmftP`PìKs

e[ìz[ìKaco

0 µilkCqj¶gedcnmfmf>tdP`k

a, e, b, c, d iKm

c, e, a, b, d kotPmI[ìlk lt^`g~ilk^`@kuitÌhìyed^ÌhìKPm pved>k

s±erqj¶3gèdcumomftP a, c, e, b, d

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N^`Pì ls>edkokuiiKm

E^`Pì ¤tPmftPb[` `P`>iko^jc

NmfiKs>s>iKkÐ^ì g~t^`cÐmot^jm

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lslpwilPmfkN

[ìN

e^ É kot^`k5¶3iKP`koiKprq`s>iKk1[ìCmfed>ss>i2 N1, N2

[ìN Í imomoircoiKs±emo>tPÊilkumIP`tdmoli

N → N1, N2 bÏâX¤ ZR~ vFLNM LM LC¨ Í È ª·¬,µivmfcu>g`siKm

(N , E,→)ilkumC^`P £T¡YTYATÏ R~Q ´ R~®¤ ZR~koNs>iKk

ltdP`[`mo>tPjk<ko^`ydePmoilkkotdPm4cuilkog~ibmfKilkC ¥ z

N → N1, N2 es>tcuksxi É kumoie 6∈ E(N)

im\^`Pìvgedcnmfmo>tPE1, E2

[ìE(N)eyi

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gt^jci ∈ 1, 2

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N e s>i4QS£j¡VXY|V¡%iKmE1, E2s>e\®lY ZR~âÖx[ì

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¥ !Xt^`c^`Pì1kuibmf>tdPE1, E2

[ìN ∈ N

ime 6∈ E(N) >s~Óberi É eXmfiKpviKPm4^`Pji[jbltdpÑ¶

g~tkomftPbkopvg`siN1, N2

[ìN

eyibs>iIpedcu^ìl^`celtcucoiKkog~tP`[edPm

E1, E2 mÌePmâ[`tP`P`<^jPìÐkoi Kmo>tPE1, E2

[ìN

ime 6∈ E(N) tP\P`tdmoiÐgedc

N(Ei; e) i ∈ 1, 2 s.]_^`Pj>^ì%Ks>lpwilPm<[jiN

moils^ìE(N(Ei; e)) = Ei ∪ e

iKmN → N(E1; e), N(E2; e) ¥ tm

E1, E2^`PjikuibKmo>tPw[ì

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A,E(N)\A

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iKkum\^`Pjizkoi KmftP[jiN(E1, e). ¥º« t>iKPm

E1, E2 E3, E4[ìl^ É koi Kmo>tP`kC[ì

N ∈ Nmoilss>ilk^ì

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N(E1; e)(N3; f) = N(E3; f)

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e1 e2 iKme3

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ime2, e3, E(N) \ e2, e3

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ilknm4^`Pìkoi KmftP©[ì

N ÕY¸QSQSY v2LJFLPOFQR+S

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F ′ Y Y ` R ] aV] f f ]32[SNR Q Uhf aV] N ^P][S f QR+S F = F ′ ∪ T^PQT

] f S 8 ] Uhf ] `ji ]aV] fEi R g ZhYXW[SR+SR Q Uhf SW8RzRY ] f aV]E(N)

Pl QW f Y Y ` R ] F aV] f f ]32[SR Q Uhf aV]N

] f S YXR i ] ` ] U Si R g ZhYXW8SNR+SRX] C] Z T:f ^ f R ] 2$Y7aXW ] aV]Da,432_Q ` Z Q f R+SNR Q U Z Q f_f '$aV] Y ZW_Q_ZW8R 4[S.4 aV] SNWY Uhf R+SRzR+S 4_^ Y QW f F ] f Si R g ZhYXW8SNR+SRX] 4 ` Q Uhf SWYXSNR Q U ^`g`g~tkutP`kw|^ji

E1, E2iKmE3, E4

kut>iKPm2

kuibKmo>tPjkw[jiN

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E(N) im1[`tPCilkum%[edP`kF >PjtP ltpwpvis>ined[jcoir[ìC[`bKtpvg~tkumftPg~tkuk$l[ji\s±evgjcotg`cu>mf

[U]>PmfiKcokuibKmo>tP E1 ∩ E3, E2 ∪ E4iKkumr^jPìzkoi Kmo>tP[ji

N iKmr[`tPvilkumC[edP`kF ip Kpvi

E1 ∩ E4, E2 ∪ E3 E2 ∩ E3, E1 ∪ E4imE2 ∩ E4, E1 ∪ E3

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iKmE3\E1, E(N)\(E3\E1)

kutPmº[jilkºkuibmf>tdP`k Ë P[jbltdpvg~tkoiNko^hbliKkoku¶yiKpviKPm

Ngedc2liKk

3kuibmf>tdP`k iKP^¸mf>s>koedPmUcoiKkog~ibmfydilpwilPmXs>iKkµpedcu^ìl^`cuk

a b im c tm N ′ ∈ N>kuko^[ji liKkC[` ltpwg~tkomf>tdP`k g~tkokul[ePmrs>ilkpeco^ìl^jcoka b iKm

c a, b, E(N) \ a, b im

a, c, E(N) \ a, c

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imE3, E(N) \ E3kotdPm[ìKkkuibmf>tdP`k[ji

N tP gedc1s±ebg`cutg`cu>KmoÑ[ì\mfcfeP`komfy|mf b, c, E(N) \ b, c ilknm

^`PìÑkuibKmo>tP¦[ìN ′ iKmgedcs>ezg`cotgjco>mfÑ¥ ¸ E1∆E3, E(N) \ (E1∆E3)

iKkum^`PìÑkoi KmftP[ìN

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NilPb[` ltpwgtdkfedPmco l^`cukoyilpwilPm

Ngec,[ìKk<[`bKtpvg~tkumftP`kxko>pwg`silk È Piim KtpwpviÒe^l^`PjiNkuibmf>tdPPì ÌhjiKyed^ÌhìN^`PìÕkuibKmo>tP ¤tcnmfiÒ[ì

N s±eÐkuibmf>tdPsKtcocuilkog~tP`[jcfe\^`Pìkoi Kmo>tP©[ì

N1

t^N2

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k] f S Y YXW ] T W aV] 2& Rk T ]

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T kotm t∗h ^`Pjik¶3i É gjcoilkukotP[ì

G∗h

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(2k+1)¶3i É gjcoilkukotPÑgt^jc

GilP^¸mf>s>koedPm

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TiKm

tP;KtP`kumoco^`mÕco l^`cukoydilpwilPmN^`Pì(2k+1)

¶3i É g`coiKkokotPth

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0, 1, . . . , 2k moils>si%^ì∀v ∈ NGh

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^`Pì2k

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[ìG∗

h − vh

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1, . . . , 2k mfilss>ir^ì∀v ∈ NG∗

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i ∈ 1, . . . , q tPko^`q`knmfmf^ì[edPjk

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vhi

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ρ1→l(thi) ¥@>PedsilpwilPmtP

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(vhi)iKm%kotm

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h \ v1, . . . , vq Ë PP`tmfi\^ìC,

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G = (V,E) TVU WY[Z \ ]j][S T = (VT , ET ) f Q U YXW i W ]jaV] a,432_Q ` Z Q f R+SNR Q U ] U2_Q T Z ] f Fl QW f Q U Y ∑h∈VT

n(G∗h) ≤ 3× n(G)− 4

][S ∑h∈VT

m(G∗h) ≤ m(G) + n(G)− 3

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m + n− 3 10052 %; :Õ= =A =,EÕ?¸=: &;>= ?,J &;=

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HeKtpwpviÑg~t>[`k

maxv∈V ∩NH′(u) w(v) t#H ′ ilknm,s.]_^jP`>^ìed^¸mfcoiacoedg`hìpecfeXmflcu>knmf>^ìg~tkokul[ePmxs>i<pvedcu|^jil^`c

u |Í iKm,eds>adtcomfh`pwi,PU]ilknmgedk KtcocuibKms>e<a^`cui ¸ rilP©g`culkoiKPmfiC^jPrKtPmfcuii É iKpvgjs>i

37% 5 h#N % Nh% !, 2 ,#

G

d

ew vw

ab

cv

a b c

ed

ÀjÁÂ¾ÃP`ÃÆ Ç °ed^hÌhìÕ^`P acfeg`hì©[ìgecomo)gt^jcwsil^ìKs4s.]es>atcumoh`pvigt^jcws>emfed>ss>iz[jiÊs>eg`s^`kacfedPj[ì ls>^ì[`tP`P`[edPjkC¨ Í ¬ Ìh`t^ì t^`k4s>iKkgt[`kkotPm4[ì

1 cutmoiÑkotPedcoq`cui[ìb[` ltpwg~tkomf>tdPSilPoKt^`g~ilk ^`,ilkumÑKaedsilpwilPm\kotdP'^jP`>^ìb[` ltpwg~tkomf>tdPSg`cuilpw lcui iKkpvedco^ìK^`cokkotdPm

viKmw ω(G) = 4

iKm4s>ercolg~tP`kui[ìsà]_eds>atdcomoh`pwiiKkum3

µi<s>ilpwpwi j ·I[`tdP`Pìsà]_eds>adtcomfh`pwiÒ[ì Í ^`P`Pj>Pàhèdp µi<s>ilpwpwi j [`tdP`Pì<^`Pwedsatcomfh`pwig~t^`c es l^js>ilc<si1gtd>[`k<pve É >pr^jp8[U]^`Pì Ks>>^ì ilP©^jmo>s>kfedPm<s>er[`bKtpwgtkumo>tPbiKPKt^`g~ilk tm

G = (V,E)^jPAacoedg`hì)mfiKs^ì

G = G1 ∗ G2 G1 = (V1, E1) G2 = (V2, E2)im

V1 ∩ V2 = v tm w : V → N^`Pì ¤tPhKmftP)[ìgtdP`[`lcoemftP

ÕY¸QSQSYmvFLM "!#CV@ J + -zL OFQR+Sa = αw(G2 −NG2 [v])

][Sa′ = αw(G2 − v)

Fl QW fαw(G) = αw|v→a′−a

(G1) + a.

ÕY¸QSQSYmvFLMMLpOFQR+Sa = ωw(G2[NG2(v)])

2l QW f ωw(G) = max(ωw(G2 − v), ωw|v→a

(G1))

4 ` Q Uhf SWYXSNR Q U Ë P e >pwpwl[`>emfiKpviKPmωw(G) ≥ ωw(G2 − v) i g`s>^jk¦tdP e

ωw(G) ≥

ωw

(G[(V1\v)∪NG2(v)]

) im ωw

(G[(V1 \ v) ∪NG2(v)]

)= ωw|v→a

(G1) ledc NG2(v)iKkum@^`P

pwt¸[`^js>iÕ[ìG[(V1\v)∪NG2(v)] È P;KtP`kul^ìlPhli ωw(G) ≥ max

(ωw(G2 − v), ωw|v→a

(G1))

È PAt^jmocoi tdPAeωw(G) = max

(ωw(G2 − v), ωw|v→a

(G1)) g`^`ko^ìÊko

Cilknm^`PjirKs>|^ji

[ìG

es>tcukrkotmC ⊆ V2 \ v

iKmw(C) ≤ ωw(G2 − v) kotm C ⊆ (V1 \ v) ∪ NG2(v)

imw(C) ≤ ωw|v→a

(G1) µilk1s>iKpvpwilk j ·iKm j coiKkog~ibKmoydilpwilPm g~il^jydilPmnKmocoi\^jmo>s>koKkIg~t^`ccleds K^`s>iKc

αw(G)im

ωw(G)[U]^`Padcfedg`hji

GilP^jmo>s>kfePmIkfe)[`bKtpwgtkumo>tP¼iKP lt^`g~ilk

T Ò]>[`Ki\ilkumnKils>sir[`tPjP`ligedc Í KilcotdPìziKm mfiKg`hedP`t ¨ Í ¬ Ë PSg`coiKP`[ ^`PìDKt^`g~i

V1, V2[`âcoedg`hì eyi Eltpwpwi

[` ltpwgtdkomo>tPSko>pwg`siG1

imG2 [jibpedcu^ìl^`c

v moilsk\|^jiG2

ilknmv^jP'adcfedg`hjibg`coiKpvilc ^`Pacoedg`hìcKtpwg`s>imÐt^)^`PìIKmftd>s>i F.Í ]_iKkumC[`coi%tdPbg`coiKP`[)^`Pì ltd^`gi KtcocuilkugtPj[edPm4r^`PìedcKmoi[ìwsà]_edcoqjcoiÑ[ìw[`bKtpvg~tkumftP>PK>[ìKPmfiwÊ^`Pjij¤iK^`>ss>i 5 Ë P leds K^`s>i

a = αw(G2 − NG2 [v])im

a′ = αw(G2 − v) g`^`ktP leds K^`s>i©co l^`cukoydilpwilPmαw|v→a′−a

(G1) ]_edg`clkbsiÊsilpwpvi tdP eαw(G) = αw|v→a′−a

(G1) + a ^ Peds tPÊed^`coejleds K^`s>ed^©g`s^`k[ìl^ É ¤td>k<s>i1g~t[`k<pe É pr^`p[U]^`P©ilP`kuilprq`sikumÌeq`s>i1ku^`c Ìhèd^ìadcfedg`hji edcoeXKmolcokumo>^ì[ìs±er[` ltpwg~tkomf>tdP

q, ? ,#

100./ bHK7 &;:¯=º? 9<=#9<7 .EÕJ9>7%E t^jk4mf^`[j>tP`k<[èdP`k limomoikuibmf>tdP©si%g`cutq`s KpviI[ì[`tpw>Pemo>tPviKmkuilk<yedco>edPmfilk tm G =

(V,E)^`P)acfeg`hì%mfiKsU^ì

G = G1 ∗G2 G1 = (V1, E1) G2 = (V2, E2)imV1 ∩ V2 = v tm

w : V → N^`Pì ¤tPmf>tdP©[jig~tP`[jlcfemf>tdP µi¦silpwpviko^`ydePm)Pjt^`k)pvtdPmocoi ltdpvpwilPm'es l^js>ilcs±eSmÌe>s>sipw>P`pr^`p [U]^`PilP`kuilprq`si

[`tpw>PèdPm ltPjPì É i eyi ^jPì[jbltdpvg~tkomftPzilPrltd^`giKkku>pwg`s>i ÕY¸QSQSYmvFLMVJ2LpOFQR+S

a1 = minu∈V1 w(u)][Sa2 = minu∈V2 w(u)

?l QW f Q U Y 5

γ w(G) =

min(γ w(G1 − v), γ w(G2 − v), a1 + a2

) f R NG1 [v] = V1

][SNG2 [v] = V2

^γ w|v→a1

(G2) f R NG1 [v] = V1

][SNG2 [v] 6= V2

^γ w|v→a2

(G1) f R NG1 [v] 6= V1][SNG2 [v] = V2

^γ w|v→0

(G1) + γ w|v→0(G2) f R NG1 [v] 6= V1

][SNG2 [v] 6= V2

4 ` Q Uhf SWYXSNR Q U edP`kIs>ig`coiKpvilcpedk G iKkum%s>e%utd>Pmf^`cui[jiG1 − v

iKmG2 − v OPÊilP`kuilprqjs>i

[`tpw>PèdPmdltP`Pji É i%pw>P`peds|ilknm,[jtP4kotmx^jPilPjkoilpCq`s>i4[`tpw>PePm KtP`Pì É i[ìG1−v kotdmN^`PilPjkoilpCq`s>i[`tpw>PèdPm ltPjPì É i[ì

G2 − v kotdmsà]^`P`tPb[U]^`P©kotpwpvim<[ìG1 − v

iKm4[ìG2 − v

edP`kCs>iw[ìl^ É lpwi ek ko D ilknmr^`PiKP`koiKprq`si[`tpw>PèdPmtKtP`Pì É ipvP`>pr^jp [ìG

es>tcukD ∩NG2(v) 6= ∅ D ∩ (V1 \ v) 6= ∅ edstcok |D ∩ (V1 \ v)| = 1

g`^`>ku^ìD

ilknmIpw>Pj>pveds iKmedP`kow(D ∩ (V1 \ v)) = a1 µi%mocotko Kpvi edkilknmIku>pw>s>ed>cuie^Ê[ìK^ É Kpvi

edP`ks>i\[jilcoPj>ilc ledk kutm D ^`P¦ilPjkoilpCq`s>iÑ[`tpw>PèdPm ltPjPì É iw[ìG D ∩ NG1(v) 6= ∅

iKmD∩NG2(v) 6= ∅ s>tdcok

D∩V1∪vilknmÕ^`P\ilP`kuilprqjs>iÐ[`tdpvPedPmKtP`Pì É i<[ji

G1

iKmD∩V2∪vilknm4^`PÊiKP`koiKprq`si[jtpvPedPm KtP`Pì É i[ì

G2 P`koγ/w(G) ≥ γ

/w|v→0

(G1) + γ/w|v→0

(G2) isà]_ed^jmfcuicmf ku D1

iKkum4^`P©ilP`kuilprq`si[`tpw>PedPm KtP`Pì É i[ìG1

imD2

ilknm4^`P)ilP`kuilprq`si[`tpw>PèdPm KtP`Pì É i,[ì

G2 es>tcuk D1∪D2\vilkumltP`Pji É i iKkumÒ^jPrilP`kuilprqjs>iÐ[`tdpvPedPmX[ì

GiKm

w(D1∪D2 \v) = w(D1)+w(D2) 7 ^ PedstPeγ/w(G) = γ

/w|v→0

(G1)+γ/w|v→0

(G2) µipjlpwi tPÊevsisilpwpviku^`yedPm4gtd^`csig`cotqjs lpwi[ìtKs>|^ji[jtpvPedPmfi evg`cuil^jydiCilknm

mfclkkopvs±edcoi;lilss>i1[`^©s>iKpvpwi j ÕY¸QSQSYmvFLMVv2LpOFQR+S

a1 = minu∈V1 w(u)][Sa2 = minu∈V2 w(u)

?l QW f Q U Y 5

γ w(G) =

min(γ w(G1 − v), γ w(G2 − v), a1 + a2

) f R NG1[v] = V1

][SNG2 [v] = V2

^γ w|v→a1

(G2) f R NG1[v] = V1

][SNG2 [v] 6= V2

^γ w|v→a2

(G1) f R NG1[v] 6= V1

][SNG2 [v] = V2

^γ w|v→0

(G1) + γ w|v→0(G2) f R NG1[v] 6= V1

][SNG2 [v] 6= V2

!@t^jcNsilkÕg`cutq`s KpviKkÒilP`kuilprqjs>ikumfedq`s>i ls>^ì<imxsilkÕ[ìK^ É g`cublK[ìlPmoilk,yedcu±edPmfiKkÒ[U]ilP`kuilprq`si[`tpw>PèdPm si\g`cotqjs lpwiÑbcoKkot^`[jcoiwko^`cs>iKk ltpwgtdkfedPmfiKkmÌedm1kotdms>iÑpjlpwi kutms>eyiKcokotP

37% 5 h#N % Nh% !, 2 ,#

g~tP`[`Kcoli[`^gjcotq`slpwi !Xt^`c4ilP`kuilprq`si[jtpvPedPm tdPewqiKkotPÊ[ì[` `P`>c^`PÊg`cotqjs lpwig`s>^jkaKP`lcoeds tm

N = N ∪ +∞iKmw : V → N

4 tm Πi : N4→ N

s>e¼g`cotui Kmo>tP [ì)s±elpwilt|tcu[`tP`P`Ki i ∈ 1, 2, 3, 4 tm

w(G,D,w, u) =

Π1(w(u))kuu ∈ D

imNG(u) ∩D 6= ∅,

Π2(w(u))kuu ∈ D

imNG(u) ∩D = ∅,

Π3(w(u))kuu 6∈ D

imNG(u) ∩D 6= ∅,

Π4(w(u))kuu 6∈ D

imNG(u) ∩D = ∅.

µi ´ R~ZT@®\½¾Ï|XÏ¡£WZ.®KÏ[U]^`P©ilP`kuilprq`siD

[eP`kG

ilknmw(G,D,w) =

∑

u∈V (G)

w(G,D,w, u).

tmCW¤Y RQ¹X¡Y TXY TXR~QZ£j ¤R ½¾Ï|Ï|¡£WZZ®lÏ¼[ìG Pjtdmf γ (

w(G) siÑgt[`kalP`Kcfeds>ku

pw>P`pr^`pG[U]^`Pkut^`kn¶DilP`kuilprqjs>i[ìkotpwpvimfkD

[ìG sµilkum ZeXK>si%[ìytc<|^jiP`tdprq`coi1[ji[`t¶

pw>Pemo>tPÑaKP`lcoeds>ko4ilknm<^`Pì%adlP`Kcfeds>kfemf>tdPw[]_iKP`koiKprq`s>iI[`tpw>PedPm gj^`>ku|^ji γw(G) = γ(w′(G)eyi

w′(u) = (w(u), w(u), 0,+∞)g~t^`c\mft^¸m

u ∈ V skl]_edamved^`koku[]_^`PjiÊaKP`lcoeds>kfemo>tP[U]ilP`kuilprqjs>ir[`tdpvPedPmmotdmÌes eyi w′(u) = (w(u),+∞, 0,+∞) iKm1[U]ilPjkoilpCq`s>i[`tpw>PedPm>P¸¶[`KgiKP`[edPm eyi w′(u) = (+∞, w(u), 0,+∞) ÕY¸QSQSYmvFLM HLpOFQR+S

a1 = γ w|v→(0,∞,∞,∞)

(G2),

a2 = γ w|v→(∞,∞,0,∞)

(G2),

a3 = γ w|v→(∞,0,∞,∞)

(G2),

a4 = γ w|v→(∞,∞,∞,0)

(G2).

l QW fγ

w(G) = γ

w|v→(a1,a2,a3,a4)(G1).

4 ` Q Uhf SWYXSNR Q U tm w′ = w|v→(a1,a2,a3,a4) tm D ^`P¼kut^`k5¶3ilPjkoilpCq`s>i\[ìV ^`g`g~tkotPjk%^ì

D ∩NG2(v) = ∅imD ∩NG1(v) 6= ∅ edP`k li ledk tP©tqjmo>iKPm

ww′(G1, D ∩ V1) = a3 +∑

u∈V1

ww(G1, D ∩ V1, u)

≤ ww|v→(∞,0,∞,∞)

(G2, (D ∩ V2) ∪ v

)+∑

u∈V1

ww

(G1, D ∩ V1, u

)

≤∑

u∈V2

ww

(G2, (D ∩ V2) ∪ v, u

)+∑

u∈V1

ww

(G1, D ∩ V1, u

)

≤ ww(G,D).

q, ? ,#

pvs±edcoilpwilPm ko D ∩NG2(v) = ∅iKmD ∩NG1(v) = ∅

eds>tcukww′(G1, D ∩ V1) ≤ ww(G,D) imko

D ∩NG2(v) 6= ∅edstcok

ww′(G1, (D ∩ V1)∪ v) ≤ ww(G,D) edPjkÐmft^`k,s>ilk ledkÐtPbtdqjmfilPmγ(w′(G1) ≤ ww(G,D) e >Pjko

γ(w′(G1) ≤ γ

(w(G) tm

Di i ∈ 1, 2, 3, 4 [ìlk4kot^jkn¶3iKP`koiKprq`silk[ìV2 ∪ v

moilsk4^ìa1 = ww|v→(0,∞,∞,∞)

(G2, D1),

a2 = ww|v→(∞,∞,0,∞)(G2, D2),

a3 = ww|v→(∞,0,∞,∞)(G2, D3),

a4 = ww|v→(∞,∞,∞,0)(G2, D4).

tmD′ ^`P©kot^jkn¶3iKP`koiKprq`si[ji

V1 ∪ v tm

D =

D′ \ v ∪D1

kov ∈ D′ iKm D′ ∩NG1(v) 6= ∅

D′ \ v ∪D2

kov ∈ D′ iKm D′ ∩NG1(v) = ∅

D′ ∪D3kov 6∈ D′ iKm D′ ∩NG1(v) 6= ∅

D′ ∪D4kov 6∈ D′ iKm D′ ∩NG1(v) = ∅

stcokγ(w(G) ≤ ww(G,D) ≤ ww′(G1, D

′) P`koγ(w(G) ≤ γ

(w′(G1)

td^jm Ktpwpvibg~t^`c\silkÑg`cotdq`s KpviKkwiKP`koiKprq`siÊknmÌedq`sibiKm ls>^ì lilkvs>iKpvpwilkrg~il^¸yilPm Kmfcui^jmo>s>koKkÐgtd^`c edsl^`silc<si1gedcfep mfcoi

γ(w

iKPÊ^¸mf>s>koedPm<sà]_edcoqjcoi1[ì[` ltpwgtdkomo>tPilPrlt^`g~ilk 100 7 &xHK= ÕM<HK7¯J91:Õ= =A µ7I;7ÐHlJ97%E

Ë P¼ilknm1pe>PmfiKPedPmIPmflcuilkukorgecIs>ines l^js@[`^P`tprq`cuitÌh`cutpvemf|^ji[U]^`Pacoedg`hìilP¼^¸mf¶s>kfedPmÑkutPedcuq`coi)[ì©[` ltpwgtdkomo>tP iKP Kt^`g~ilk Ë P'kfedmv[j5o|^jiÊ[eP`kvsiEedkw[ìKkvacoedg`hìKkgedc"ZedmokÐs>i1gjcotq`slpwi1koi ltdpvg~tcumoi1q`>iKP'edc>silkum4K|^jyeds>iKPm4ed^Êgjcotq`slpwi $31) -

edk1[edP`k1siCedk1adlP`Kcfeds s>iCg`cotqjs lpwirkuilprq`sirg`s^`k ltpwg`si É i g`^`>ku^ìÑkoG = G1 ∗ G2

χ(G)Pìg~il^jmCgedksKmfcui[`K[`^`mtZeXK>s>iKpviKPm[jilkCkuts>^¸mf>tdP`k[jilkCkut^`k5¶3g`cutq`s KpviKkCku^`c

G1

iKmG2 Ò]es>atcumoh`pvi edsl^`silcoe¼s>i)P`tprqjcoiEÌh`cutpemf>^ì)g~tP`[`Kco)[ìKkvkut^`kn¶Dacfeg`hìlkjltdcocoiKkog~tP`[ePmok

ed^ É P`t|il^j[`kI[ìsà]_edcoqjcoi[ìC[` ltpwg~tkomf>tdPÊiKP=lt^jgiKkIilPjcfeXK>P` [` P`@vs±ewkoi KmftP ¸7 gt^jc^`Pì;Kts>sibmf>tdP¼[ìC¤tPmftP`k1[ì\g~t>[jk tm G ^`Pacfeg`hì iKm T kutP¦edcoqjcoi\[ì\[` ltpwg~tkomf>tdPilPrltd^`giKk es l^js>1gedci É iKpvgjs>iilP©^jmfs>kfedPm<s.]edsatcumfhjpvi4s>Pèdcoi%[ji edh`s>hèd^`k¨ edh··¬ Ë PÌh`tkomÒedcuq`mocfedcoilpwilPmº^`P\P`t|il^`[

r[jiÐsà]_edcoqjcoiÐÛ]tPÑg`coiKP`[`coe KtpwpvixcfeXK>PìÐ[ji

Tgt^jcâs>iÐcuilknmfi

[ì limomoikuibmf>tdP tdmocoiedsatcomfh`pwimfcfeyeds>s>iKcfeCcubK^`cokuyiKpviKPmIku^`csà]_edcoq`cuiT

iKP`cfeXK>PìKPr Ë Pbcfedg`g~ilss>i4^ì%g~t^`c,^jPbP`t¸iK^`[

h[ìT G∗

h

iKkum<siacoedg`hì edcoeXKmolcokumo>^ì KtcocuilkugtPj[edPmed^wP`t|il^`[

h[ìT Gh

iKkumNs>i<acoedg`hì Ktcocuilkog~tP`[èdPmÐed^Ñkut^`kn¶Dacfeg`hìilP`coeXlPìlPh iKmNko h 6= r

vh

iKkums.]_^`Pj>^ìrpvedco^ìK^`ccKtpvpC^`PG∗

h

imG∗

h′ t# h′ ilkumsiCglcui\[ìh

[èdP`kT tdmfi\^ì

Gr = G Ë Pvg`culkuilPmfi4P`tdmocoiedsatcomfh`pwixs>ea^`cui j ² e%yeds>[`mo,ilknmNqedkoKiko^`cÕs>iKkÒs>iKpvpwilk ¸ %imj ² µiKkNPjtdmÌemo>tPjkÒko^`ydePmoilkÕkotdPmN^jmo>s>koKilkâmot^jmNed^ws>tPja1[ìs±e%koi Kmo>tP tm C ^jPì ltstcfemo>tPw¶DgtdP`[`lculi[ì

G = (V,E)imV ′ ⊆ V Ë P©P`tdmoigèdc

CV ′

s>eCltss>i KmftPb[ì1mftd^`ks>ilkiKP`koiKprq`s>iKk

37% 5 h#N % Nh% !, 2 ,#

"!#(T )$%'&)(+*-,/.1032546()78(+,:98,;98*-<-=?>A@B=C)DE&)DF=?%G,H%I<-=?J8@B,-C

T98,

G KL =?()&)DF,M.NF,%1=?>O78(+,;<+P8(+=?>M46&)DFQRJ1,;98,GSTRUWV

X VY Z V %1=R,HJ19 h98,

T K 98,-C\[],HJ8DENENF,-C"^ N_4 (`4?<HDE%1, K =?a k,-C)&NF,%1=?>;78(+,O98,b1NFC

98,h,H&

h1, h2, . . . , hk

C+=?%'&"C+,-C\b1NFCc

h 6= r d1e Y cDh ← ∅X VY b

Sf0 g n hid Y6f

Dh ← Dh ∪ (kjBlm'

(h, b, k,Dh1 , Dh2 , . . . , Dhk)− b, b)n X VY

c o Y6f Z VY f'Y pjBl1m' (r, 0, k,Dh1 , Dh2 , . . . , Dhk)n c

n X VYn rq6stP8(+=?>M46&)DF<-u"J8>;7,H(Hv

kjBlm'(h, b, k,Dh1 , Dh2 , . . . , Dhk

)STRUWV c

k = 0 d1e Y cc B_w∗ : V ∗

h → N&+,HNQRJ1,

∀v ∈ V ∗h \ vh K w∗(v) = 1

,H& K C)D h 6= r K w(vh) = bY6f Z VY fRYχw∗(G∗

h)c c B_

m← +∞X VY cSWf

0 g 2n hwd YxfX VY iSWf

1 g k hwd Yxfc B

ai ← min

a′ : ∃b′&+,HNpQ'J1,

(a′, b′) ∈ Dhi

,H&a′ + b′ ≤ c

n X VYc

w∗ : V ∗h → N

&+,HNQ'J1,@=?J8(&+=?J8&i ∈ 1, 2, . . . , k K w∗(vhi

) = ai K@B=?J8(&+=?J8&v ∈ V ∗

h \ vh, vh1 , . . . vhk K w∗(v) = 1

,H& K C+D h 6= r K w(vh) = b

m← min

(m,max

(c, χw∗(G∗

h)))

n X VYY6f Z VY fRY

mn c n rq6stP8(+=?> L @8NEDE&Hv

ÀjÁÂ¾Ã`ÃÆ©Ç s>atcumoh`pvi edsl^`s>edPm<s>i1Pjtprq`cui Ìh`cotpvemo>^`i

q, ? ,#

kumfedq`silkÐ[ìIs±esKts>tdcfemo>tPC

^`,ltPmo>ilPme^bpvtP`kN^`PbkotpwpwiKmx[ìV ′ ]_iKkum4C[`coi

CV ′ = 〈S ∈C : S ∩ V ′ 6= ∅〉 Ë PPjtdmfiwgedc

D(G,w, V ′)s.]_iKP`koiKprq`s>iÑ[ìÑmftd^`ks>ilknltd^`g`s>iKk

(a, b) ∈ N × NmfiKs>krÛ]>sÕÓ em^`PìEltds>tcoemftPx

g~tP`[`KcoKiC

[ìG

eyi a + b

lt^js>il^jcokrim|CV ′ | = a Ë P

ytdmCZeXK>silpwilPm\^ì)ko(a, b) ∈ D(G,w, V ′) eds>tcuk (a′, b) ∈ D(G,w, V ′)

gtd^`cCmftd^jma′ > aiKm

(a, b′) ∈ D(G,w, V ′)gtd^`c\mft^jm

b′ > b Ë P'[`mvÛ]^`Pult^`gjs>i(a, b) ∈ D(G,w, V ′)

ilknmQZ@ZQS£WNko

(a − 1, b) 6∈ D(G,w, V ′)im

(a, b − 1) 6∈ D(G,w, V ′) tdmfiv^ìvgtd^`c Ìhed^ìltd^`g`s>i1pw>Pj>pveds

(a, b)[ìD(G,w, V ′) tPÊe

a ≤ χw(G)imb ≤ χw(G)

gedcnmf>c[ìpedPmfilPèdPm g~t^`cEÌhèd^ìP`t¸iK^`[h 6= r tdP eg`giKs>si¼sà]ilP`kuilprqjs>i

D(h) =D(Gh− vh, w,NGh

(vh))s>iD

Y|@®KY¸Q¹@WZYw[ìh t#bg~t^`cxmft^jm

v ∈ Vh \ vh w(v) = 1 tdmocoiedsatcomfh`pwi^¸mf>s>kuizs±e¼g`cutacoedpvpvemo>tP[¸Ó¸Pèdpv^ìbiKmjes l^js>ib[ìKkvP`t|il^j[`ks>ecoeXlPì gt^jcÌhed^ìP`t|il^`[

h 6= r[ji

T s>i D ¶DilP`kuilprqjs>i[ìh

[jilg`^`ksilkD¶3iKP`koiKprq`silk4[ìmot^`k4koiKk sk ^

Pes sFedsl^`si χ(G)[ìlgj^`>ks>iKk

D¶3iKP`koiKprq`silk[ìlksk<[ìs±e\cfe l>Pji

r µivsilpwpvi\ku^`yedPmi É gtdkoiwsà]tgKcfemo>tP¦)i É bK^jmfiKcrg~t^`c Ìhed^ìvP`t¸iK^`[S[jivs.]edcuq`coiÑ[ìw[` ¶ltdpvg~tkomftP g~t^`c edsl^`silckutP

D¶DilP`kuilprq`si¼iKmzkotP P`tdprq`coiqÌh`cotpvemo>^ì ilP ^jmo>s>kfePmvs>iKk

D¶DilP`kuilprq`silkÕ[ìkoiKk `s>k tm h ^`PvPjt¸iK^`[v[ji

T tdm G∗h = (V ∗

h , E∗h)

s>i<acoedg`hì ledcfeXmfKco>knmf|^jiltdcocoiKkog~tP`[ePm

h iKmkutmGh = (Vh, Eh)

s>i%adcfedg`hji ltcucoilkugtdP`[edPmed^)kot^jkn¶3adcfedg`hjiiKP`cfe l>PjilPh tm h1, h2, . . . , hk

silk9s>kÕ[ìh[eP`k

T 7 s>tcukGhi

iKkumxs>iacfeg`hì ltcucoiKkog~tP`[edPm,ed^kut^`k5¶edcuq`coiilPjcfeXK>PìlP

hi

iKmvhi

ilknmIs.]_^`Pj>^ì1pvedco^ìK^`cilPrltpwpr^`P©G∗

h

iKmGhi

!@t^`cku>pwg`s> ilcs>iKk%P`tdmfemftP`k[edP`k%siCsilpwpvi j iKm1koezg`coiK^jyi tPP`tmfilcoe

Gi = Ghi= (Vi, Ei) vi = vhi

imG∗ = G∗

h = (V ∗, E∗) ºÍ tpwpviw>sxemfz[`mCgjcobKl[ìKpvpwilPm Gh = G∗ ∗ G1 ∗ . . . ∗ Gk tmw : Vh → N

^`Pì ¤tPhKmftP)[ìgtd>[`k ÕY¸QSQSY vFLMFL - Q T W SQ T S i ∈ 1, . . . k ^ f QR+S (ai, bi) ∈ D(Gi − vi, w,NGi

(vi))COFQR+S

C 8 ] Uhf ] `ji ]KaV] SQ T S] f ] f 2_Q QWYXSNR Q Uhf w g Z Q U a,4[W34_] f C aV]Gh

YX]32|CVi\vi| = ai + bi

][S|CNGi

(vi)| = ai

2l QW f

minC∈C|C| = max

(χw∗(G∗), max

i∈1,2,...,k(ai + bi)

)

Qw∗ : V ∗ → N

S]&Pk T ]sZ Q T W SQ T S i ∈ 1, 2, . . . , k, w∗(vi) = ai

^s][SPZ Q T WDSQ T S v ∈ V ∗ \v1, v2, . . . , vk

^w∗(v) = w(v)

4 ` Q Uhf SWYXSNR Q U Ë P¼cfedgjgiKs>s>i^ìrkuC

ilknm%^`PjiCltstcfemf>tdP©g~tP`[jlcoKir[ìG iKm V ′ ⊆ V es>tcuk

CV ′

ilknmN[` Pj ltpwpvi,s>iÐpr^jsmf ¶3iKP`koiKprq`s>ix[ìlkÕiKP`koiKprq`s>iKkÕkumÌeq`s>iKkÒ[ìC

^``tPmÕ^`PìPmoilcukoibmftPP`tP)y|>[ìeydib

V ′ !Xt^`c<mft^¸mi ∈ 1, . . . , k kotdm Ai = NGi

(vi)iKmBi = Vi \NGi

[vi] Ë P pvtdPmocoi^ìSmot^jmfiKkÊsilkltstcfemo>tPjkC ∈ C

ltPmfilP`PjilPmed^ pvtd>P`kmax

(χw∗(G∗),

maxi∈1,2,...,k(ai + bi)) iKP`koiKprq`silkNkumÌeq`s>iKk !@t^jcÕmot^jm i ∈ 1, . . . k ai + bi = |CVi\vi| ≤ |C|

scoiKkumfig`cut^jydilc^ìχw∗(G∗) ≤ |C| !@t^`cjliKs±e tPultPjkumfcu^`mv^`Pìrltds>tcoemftP gtPj[`lculi

C∗ [ì(G∗, w∗)

[ìlgj^`>kCVh\

Ski=1 Bi

!Xt^`cvmot^jmS ∈ CVh\

Ski=1 Bi

kutmS∗ ⊆ V ∗ s.]_iKP`koiKprq`si

yKco ¾edPmS∗ ∩ (V ∗ \ v1, . . . vk) = S ∩ (V ∗ \ v1, . . . vk) im\gtd^`cmot^jm

i ∈ 1, 2, . . . , k vi ∈ S∗ kuIimbkoiK^`s>iKpviKPmzko

S ∩ Ai 6= ∅ tdmotP`kz^ìS∗ ilknm)^`P ilPjkoilpCq`s>i¼knmÌedqjs>i[ji

G∗ tmC∗ s>ivpr^`smf¶DilP`kuilprqjs>iÑ[ìmot^`kCsilkriKP`koiKprq`silk

S∗ ºÍ s±edcoiKpviKPm|C∗| ≤ |C| izgjs>^`k

C∗

37% 5 h#N % Nh% !, 2 ,#

ilknmz^`Pji=ltstcfemf>tdPw∗ ¶3g~tP`[jlcoKi[ì

G∗ g`^`>ku^ìg~t^`cwmft^jmi |C∗

vi| = |CAi

| = ai P`kominC∈C |C| ≥ max

(χw∗(G∗),maxi∈1,2,...,k(ai + bi)

) sÒcoiKkumfivb[`lpwtPmfcuilc1s.]_Kaeds>mf !@td^`c lils>ebtP ltP`knmfcu^`m^`Pji Kts>tcoemftP

w¶3g~tP`[`Kcoli

C[ì

Gh

moilss>iI^ìC ∈ C

iKm|C| = max

(χw∗(G∗),maxi∈1,2,...,k(ai + bi)

) tm C∗ ^`Pì ltstcfemo>tPw∗ ¶3g~tP`[`KcoliÑpw>P`pr^`p [ì

G∗ imgt^jcmot^jmi kutm Ci

^`Pìjltstcfemf>tdP¼[ìGi − vi

moils>si\^ì|Ci| = ai + bi

im|Ci

Ai| = ai Ë P ltP`knmfcu^`m

C[ìs±erZeltPku^`yedPmoi edP`krs>eÊg`cuilpw lcuimÌedg~i tP g`cuilP`[ ^`PSilP`kuilprq`si

kumfedq`siS∗ ∈ C∗ iKm@gtd^`cµmft^jm

i ∈ 1, 2, . . . , k ko vi ∈ S∗ tPg`coiKP`[C^`PCilP`kuilprq`sixknmÌedqjs>iSi

[eP`kCi

Ai ku>P`tPwtPwg`coiKP`[

Si[edP`k

Ci \CiAi ]>sPì4coiKkumfied^K^`PiKP`koiKprq`siknmÌedq`si tPvg`coiKP`[

Si = ∅ Ë Peut^¸mfiC

sà]ilP`kuilprq`siS = S∗ \ v1, . . . vk ∪

⋃k

i=1 Si Ë PÊP`tdmfi^ì

Silknm1^`PÊilP`kuilprq`si

kumfedq`si Ë PzculgKmoi liKmumfi%tdgKcfemo>tPu^jkoÛ]_ Ki%^ìC∗ kotm,y|>[ì Í tpwpwi

w∗(vi) = ai = |CiAi| >sPìcoilknmfied^K^`P©ilP`kuilprq`si[edP`k

Si ∈ Cimoilsµ^ì

Si ∩ Ai 6= ∅\s±e P©[ìs±ergjcoilpw Kcoi1mÌedg~i

edP`k<s>eC[ìK^ É lpwi1Kmfedg~i mfedPmÛ]_si É kumoi1^`Pi ∈ 1, 2, . . . , k

moils|^jiC i 6= ∅ tPbgjcoilPj[ g~t^`cmft^jm

i ∈ 1, 2, . . . , k ^`PilP`kuilprq`siCknmÌedq`siSi ∈ Ci

kuCi 6= ∅ ko>PjtPtPg`coiKP`[

Si = ∅ Ë P©eut^jmoiC

sà]ilP`kuilprqjs>iS =

⋃k

i=1 Si Ë P)P`tmfi%^ì

SiKkum^`P)iKP`koiKprq`s>i1knmÌedqjs>i sUPU]ilkum4gek

[` * K>si%[ì1ytc<^ìC

ilkum4^`PjinKts>tdcfemo>tPw¶3g~tP`[`KcoKi1[ì

Gh im^ìC ∈ C B s>e PC[ìNs±eg`cuilpw lcuiÕKmÌegi C e

χw∗(G∗)ilPjkoilpCq`s>iKkºkumfedq`silk imºg~t^`cmftd^jm i ∈ 1, . . . , k

Cie

max(0, |Ci \ Ci

Ai| − |C∗ \ C∗

vi|)

= max(0, bi − χw∗(G∗) + ai

) ilPjkoilpCq`s>iKkCkumfedq`silkCcuilk5¶mÌePmok V stcokNtPze ut^jmfi4[edPjk,s±e[ìl^ É lpwimÌedg~i

max(0,−χw∗(G∗) + maxi∈1,2,...,k(ai + bi)

)

ilPjkoilpCq`s>iKkknmÌedq`silkC h >Pjko Ñs>e<P

|C| = max(χw∗(G∗),maxi∈1,2,...,k(ai + bi)

) stcok

χw(Gh)iKkums>ipw>P`pr^`pG[ì

max(χw∗(G∗

h),maxi∈1,2,...,k(ai + bi)) gedcupvUmot^`ks>iKk

Ìh`t É g~tkoku>q`silkÒ[ìKkk¶3^`gjs>iKm (

(a1, b1), (a2, b2), . . . , (ak, bk)) [eP`k

D(h1)×D(h2)× . . .×D(hk) !@td^`cmot^jmc ∈ N kotdm tc s>i

k¶3^jg`s>immoils^ìrg~t^`cmft^jm

i ∈ 1, 2, . . . k (ai, bi)ilkumI^`P=Kt^`g`si

[ì(a, b) ∈ D(hi) : a+ b ≤ c

eydibs>i1g`s^`k<gimfmag~tkoku>q`si Ë Pbytm ZeXls>iKpviKPm ku t ilknmI^jP

k¶D^`g`siKm (

(a1, b1), (a2, b2), . . . , (ak, bk)) iKm

c = maxi(ai + bi) eds>tcukxsi Ìhjt É [ìtc

iKkumÐed^zpwt>P`ked^`kukoxqtPS|^jizs>i Ìh`t É [ì

t Ò]_eds>atdcomoh`pwijleds K^`s>iχw(Gh)

ilPgjcoilPèdPm\s>ivpvP`>pr^jp gedcupvmftd^`ks>ilk

k¶3^`gjs>iKm

tc c ∈ 0, 1, . . . , 2n µi%s>iKpvpwigjcobKl[ìKPmpvtPmocfedmPKtpvpwilPmPedsl^`silcNs>iIPjtprq`cui Ìh`cutpemf>^ìgtdP`[`lcu%[ìGh[ìKg`^`>k1silk

D¶DilP`kuilprqjs>ilk1[jilk s>kI[ì

h h iKkum^`PP`t|il^j[¦[jlcoiKPm[ìr tP[`tmcedsl^`silc%si

D¶DilP`kuilprq`si1[ì

h isilpwpvi%ku^`yedPmP`td^`kpvtPmocoi ltdpvpwilPm liKs±e\g~il^jmpKmocoi Zedm ÕY¸QSQSYuv2LM,'FL OFQR+S

G = (V,E) TeU WY[Z \ ]_^ v ∈ V][Sw : V \ v → N TVU ]Q U 2[SNR Q U aV]

Z QRa f OFQR+S b ∈ N][Sw′ : V → N

S]& f k T ] w′(v) = w(v)Z Q T W SQ T S v ∈ V \ v ^P][S w′(v) = b

l QW f (χw′(G)− b, b) ∈ D(G− v, w,N(v))

^][S(χw′(G)− b− 1, b) 6∈ D(G− v, w,N(v))

4 ` Q Uhf SWYXSNR Q U tm C ^`Pìsltstcfemf>tdP

w′ ¶DgtPj[`lculiCpw>Pj>pr^`p8[ìG stcok

C ′ = 〈S \ v :S ∈ C〉

ilkumÐ^`PìcKts>tcoemftPw¶DgtPj[`lculiI[ji

G im<ered^pvtP`kbiKP`koiKprq`silkÐknmÌedq`silkxPU]eÓedPmgedk

[ìkotdpvpwiKmok<ilPrltpwpr^`P©eyi N(v) >Pjko

(χw′(G)− b, b) ∈ D(G− v, w,N(v)) ]>si É >knmÌedmX^`Pì ltstcfemo>tPw¶3g~tP`[`Kcoli

C[ìG−v

moilss>iÕ^ì|C| < χw′(G)

eÓedPmâed^Cpwt>PjkbiKP`koiKprq`s>iKkxknmÌedqjs>ilkÒP]eÓePmxgedkÕ[ìkutpvpwiKmokÕilPjltpwpr^`Pweyi

N(v) eds>tdcokÒs>i<pC^`smo¶3iKP`koiKprq`si

q, ? ,#

CtqjmfiKP^¼ilPe ut^jmÌePm

vvmot^`kIsilk4ilP`kuilprqjs>ilkIkumÌeq`s>iKkIPU]_eÓedPm1gekI[ìkotpwpvimfk4ilPmKtpvpC^`P

eyi N(v) koiKcfedm<^`Pì ltds>tcoemftP

w′ ¶3g~tP`[`Kcoli[ìG Ki|^jU>pwg`s>^ìI^`PjinKtPmfcoed[` mftP

edPmoilPePmPjt^`kkfeytdP`k ltpwpviKPm edsl^`silc%silkcKt^`g`silkpvP`>pved^ É [j^D¶3ilPjkoilpCq`s>ir[ji

h[ìKg`^`>kχw|vh→b

(Gh) b = 0, 1, . . . , n t^`k4eytP`k[`tdP%mft^¸m li^`µilknmP`bKilkokoed>cuiCrs>erg`coiK^jyi[ìsà]_edsatcomfh`pwi »ÏR¾¡¿|QSYvFLM )?L 8 Y QW8R+S\ ` ] / +0()# *0- +=ZW ] U aD] U ] U SNW4_] TeU W$Y[Z \ ] G ][S f Q UYXW i W ]saV]ta,432_Q ` Z Q f R+SNR Q U ] U 2_Q T Z ] f T ][S 2$Y 2 T ] ] U Q `ji W ] 2_\VW Q ` YXSNRk T ]saV] G 4 ` Q Uhf SWYXSNR Q U ej¤tPmftP/+,( $%

(h, b, . . .)coiKmot^`cuPì

χw(Gh) eyi w(v) = 1g~t^`c

mftd^jmv

[edPjkVh \ vh

iKm ko h 6= r w(vh) = b ilP^jmfs>kfedPm1si\s>ilpwpwi j )ims±ebcoiKpedcu^ìko^jyedPmÑs±e¼g`cuil^jydiÊ[j^'silpwpvi j !Xt^`c\mft^¸m

h 6= r s±e¼q~t^Ks>ibg`cu>PK>gedsi leds K^`s>iDh

|^jltdPmo>iKPm<mot^jmoilk<s>iKkÐgedcoiKk<pvP`>pr^jp#[`^

D¶3iKP`koiKprq`s>iI[ì

h ^jmfs>kfedPmxs>i%silpwpvi j ² h ^ `Peds sà]_edsatcomfh`pwipedsl^`si%s>i1P`tdprq`coi Ìh`cutpvemf|^ji1[ìGr = G

Ò]edsatcumfhjpvi leds K^`s>i4s>iP`tprq`cui Ìhjcotpvemf^ìI[`âcfedgjhìI[`tdP`P`%iKPzilPmfculi ilPb^¸mf>s>koedPmÕilPkotd^`kn¶Dcot^jmo>Pìw^`Pìwg`cuteKl[`^`cui leds K^`s±edPmsivP`tdprq`coi Ìh`cutpvemf|^jivg~tP`[`Kcow[ìlkacfeg`hìlknecfeX ¶mfKcokumf^ìlk sµPU]_iKkum4gedk[` *jl>si%[ìpwt¸[` ilc,sà]_eds>atdcomoh`pwig~t^`c<Û]>sFleds K^`s>i1iKPÊpjlpwi%mfiKpvgjk^`Pìpltds>tcoemftP\[ìmÌeds>s>i,pvP`>pr^jp ko¾tPw^jmfs>koi<ilPwkotd^`kn¶Dcot^jmo>Pì4^`Pì4g`cuteKl[`^`cui leds K^`s±ePmÕ^jPìltds>tcoemftPzg~tP`[`Kcolipw>P`pr^`pG[ìlkacoedg`hìKk edcoeXKmolcokumo>^ìlk »ÏR¾¡¿|QSYmv2LM FL OR 8 Y QW8R+S\ ` ] T SR R f 4 Z Q T W 2$Y 2 T ][W ] U Q `ji W ] 2_\VW Q ` YXSNRk T ] Z Q U a,4[W4taV] f WY[Z \ ] f 2$YXWY,2[S 4[W8R f SRk T ] f aV] Yoa,432_Q ` Z Q f R+SNR Q U ] U 2_Q T Z ] f @ TVU S] ` Z f aV]

f(n,m)^ Y QW f ]

S] ` Z f SQS YHaV] 8 Y QW8R+S\ ` ]/+,(#*,- + ] f S ] U O(n3 · f(n,m))

4 ` Q Uhf SWYXSNR Q U Í iKmÑes>atcumoh`pvi\i É bl^¸mfiO(n2)

¤t>ksà]_eds>adtcomfh`pwirg~t^`cCsiP`tdprq`coi Ìh`cotdpe¶mf^ìrg~tP`[`Kcorg~t^`cImot^`k%silkacfedgjhìlk ledcfeXmfKco>knmf|^jilk%[eP`k%s±ev[`bKtpvg~tkumftPiKPmlt^jgiKk imsà]_edcuq`coi%[ji1[`bKtpvg~tkumftPziKPrKt^`g~ilkg~tkokl[ì

O(n)P`t¸iK^`[`k i1cuilknmfigiK^jm KmocoicZedm<ilPbmoilpwg`k

O(n3)ilPg`co ¶leds K^`s±edPm

mina′ : ∃b′mfilsµ^ì

(a′, b′) ∈ Dh

ima′+b′ ≤ c

g~t^`cmft^jm@P`t|il^`[h[ì

TiKmÒg~t^`cºmot^jm

c ∈ 0, 1, . . . , 2n È P;KtP`koK^ìlPKi s>ixmfilpwg`kºmotdmÌes¸iKkumÕilPO(n3 ·f(n,m))

zY|QS£`¡UVYL ]cS] ` Z f Z ] T S 2[SW ] Y ` 4& R QW4 ] U O(n2 · f(n,m))Z Q T W ] f Q U 2[SR Q Uhf f S]& ] f k T ]

f(a+a′, b+b′) ≥ f(a, b)+f(a′, b′)Z Q T W a, a′, b, b′ ] U SNR ][W f/f SW8R 2[S] ` ] U SZ Q f R+SR f sY Uhf 2_] 2$Y f ]S] ` Z f Z Q T Wt] 1 432 T S][W 8 Y QW8R+S\ ` ] Z Q T W ] U Q `ji W ]<2_\VW Q ` YXSNRk T ]PZ Q U a,4[W4 f[T W SQ T:f ] f WY[Z \ ] fZW ] ` R ][W f ] f S ] U O(f(n,m))

]& Y'] f S ] 2$Y f ZhYXWt] 1 ] ` Z ] f R f(n,m) = n+m

"!#%$'&)(*$,+.-0/.-1$'2435-76089$,-12:/;< !=/>!?-12:/>@9$A5B"-0/.CD(E,@GFH3"-1C2I"CKJL$'2:/.M'$'N0O"C%@?PQJ$'2ER'STVUXWYNZ+%[\,M'-0/[]B"2"C),B5/[email protected]"!#%$'&)(H$'+.-7/>-1$,2^M'!2"!%@.,N1-0+>,2:/_N1`"!%#$'&I(H$'+9-0/>-0$'2^&)$a"B"[email protected]*!M:,N0C%&)C%2:/_3E,+9!%C4+9B"@N1C+edf,&)-1N1N0C%+G3"-76g(E,@9/.-0/.-0h'C+%Uikjml nporq%s*tuowvyxz|~%v ua2urb%Hf*",2"+B"2^M'@>,(5"C

G = (V,E)C+9/_B"25C(E,@/>-7/>-1$,2L"C

VC%2u5C%BA^C%2"+9C%&35N1C%+

V1C/V2

/.C%N1+XB"C|V1| ≥ 1

c|V2| ≥ 1

c∃W1 ⊆ V1

c∃W2 ⊆ V2

C/?N1C++.C%B5N1C%+,@9/>C+",2"+GC%2:/>@9C

V1C/V2

+.$'2:/e/.$'B5/>C+N0C%+e@./.C%+eC2:/>@.CW1

C=/W2

C/G/>$'B5/.C%+GN1C%+,@.=/>C+eC%2:/>@9CV1 \W1

C/V2 \W2

UAOEM'[email protected]"U7yC2L5$'2"2"CB"2C=AC%&)("N0C,U

f

h

g

ea

b

c

d

V1 V2

W W1 2

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2½3"-76789$'-12:/C%+/L¾r¿ÁÀEÂ +9-|V1| = 1

$'B|V2| = 1

U?Ã2 (HC%B5/L2"$,/.C%@XB"CÄ/.$'B5/.C%+^N0C%+3"-Å6(E,@/>-7/>-1$,2"+V1, V2

h'C%#|V1| = 1

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X, V \XC+9/åB"2 3"-Å6089$'-02:/U$,B5/å#%$'&)&)C#,[]C%+/áN1Ce#%,+(*$,B"@üN1C+ü#%$,B"(*C+%c

X(HC%B5/å=/>@9CDB"24&I$a"B"N0C2"$'2/>@9-0ha-bNºC/

X, V \XB"2L35-76089$,-12://[email protected],NôUFZ,3"-1C2`"CJL$'2:/>M'$,N0OEC@e&)$'2:/>@9!KE,25+e+> /.! %+9C_XB"CNbdf,&)-1N1N0CD"C%+3"-76789$'-12:/>+áC%+/D!%M',N1C&IC2X/B"2"C_df,&)-1N0N1C?3"-76g(E,@/>-0/.-0h,C,U

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ikj7d eýgfgyxoVtuv ih_j ff_z k t^|kGfV_zÃ2«h&)$'2:/>@9C%@B"2"C dÁ$'@/>C @.CNb/.-1$'2uC2X/[email protected])N1C+y35-76089$,-12:/>+_C=/N0C%+&I$a"B"N0C%+cXB"-å+.C%@.df,-7/>C4(E@N1C3"-1,-1+D5C N14/.@>,25+9dÁ$'@9& /.-1$'2 5C M'@.,(""C"!=OE2"-0C E2"+N0C(E,@>M'@>,(5"C+.B"-7h,2X/%UmlC/9/>C@9C%N1/>-0$'2c"$'252"!%C(E,@N0Ce/>"!$'@7 &ICD5UTnAc,2"$'B"+(HC%@9&IC=/./>@._"CD+.-0&I("N0-0OEC@<N1("@.CB5h'C?"CD@.!+.B"N7/w/.+"!ö87o#%$,2"2XB"++.B5@`N1C+ 3"-Å6089$'-02X/.+%cå#%$,&I&)C^N1«("@9$'("@.-0!/.!^"CÆ35-76Y(",@9/.-0/>-7ha-0/>!cüC/`2"$'B"+`+9C%@hA-0@>kC2 "!%&)$'2:/>@9C%@I"C2"$'BAh'C%N0N1C+%cEC%2+.C+9C%@9h,2:/"C%+@9!%+.B5N0/wÜ/>+"!ö87o4#$'2"2XB"+?(*$'B5@?N1"!%#$'&)(*$'+9-0/.-1$'2 &)$A5B"Nb,[email protected]`ÇÈa¾ÜÉqpømÀ5°ÁH ËÈsrºÈAfËÈAÂ?PutAC%-wv?xTüº[\B"2 M'@.,(""C

G = (V,E)h,C%# #%$'&)&)C+.B5("(*$,@9/B"2+.$,B"+6gC%2"+9C%&3"N0C

W ⊆ VC+9/üN0CGM,@>,("5Ch'C#eN1Cå&)%&)CGC2"+.C&3"N0CG"CG+.$'&)&)C/>+c°C/B"2+.$,&I&)C/

xC+9/,Ü8.#%C%2:/yoyB52I+9$'&I&)C/

y+.-Z+.$'-7/

x, y ∈ EC/x, y

2"C#w5Ch,B"#w"C(E,+Wc:+.$'-7/

x, y 6∈ EC/

x, y#w"Ch,B"#w5C

WUAøCM,@>,("5CK$'35/.C%2XB +.C@>25$,/>!

GW U!ta$'-0/áNbyÇÈX¾ÉqpømÀ5°ÁH ËÈ¿HHfÎVÀ!zZÈ

[]B"2`M'@.,(""CG,B +9$'&)&IC=/

vcX2"$,/.!

Gv caN1C?M'@.,(""C

GN(v)− vUaOEM'B"@9CK"Uny5$'2"2"CB"2`C=AC&I("N0C[]B"2"C(HC%@.&B5/w/.-1$'2"C_h,$'-1+9-12E,M,C,U

b

d

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V1, V2KH13DOQ@CRE\`:?[]I@CI@/1JFHD

VD]Iv ∈ V1

c`BE/[LV1, V2

DNL]I;KH1GOQ@CRTSU/@C1+IFHDG

L]@yD]ILD]KXBED]>D]1+IL]@V2

DNL]IKH1>/QF?KXBEDiFHDG

v c]>/1`L]I[:?I*@/1mctA$'-7/

V1, V2B52"CK3"-Å6Y(E@9/>-7/>-0$'24"C

GC/+.$'-7/

v ∈ V1

UXN1$'@9+%cV2

C%+/DB"2 &)$a"B"N1C"CG

v +.-åC/y+9C%B"N0C%&)C%2:/+9-(*$,B"@_/>$'B5/w ∈ V1 \ v

cV2 ⊆ NG

v(w)$,B

V2 ∩ NGv(w) = ∅

UKC("N0B"+%cV1, V2

C%+9/eB"2 3"-76789$'-12:/á"CG+.-*C/D+9C%B"N0C%&)C%2:/D+9-º(H$'B"@G/.$'B5/

w ∈ V1 \ vcV2 ∩NG(w) =

V2 ∩NG(v)$'B

V2 ∩NG(w) = V2 \NG(v)U

tAB"("(H$'+9$'2"+XB"CvC+9/ ,Ü8.,#C%2:/io

wUV2 ⊆ NG

v(w)+9-üC/+9C%B"N0C%&)C%2:/y+.-ü(H$'B"@K/.$'B5/

u ∈ V2

cuC+9/_Ü8.,#%C2:/_o

vC/wcZ$'B

u2[]C%+9/,Ü8.,#C%2:/o ,B"#B"2Æ"C

vC/wU+KN1$,@.+%c

V2 ⊆ NGv(w) ⇐⇒

5 3

V2 ∩NG(w) = V2 ∩NG(v)U`KCK("N1B"+c

V2 ∩NGv(w) = ∅

+.-*C/+.C%B5N1C%&)C%2:/D+9-º(H$'B"@á/>$'B5/u ∈ V2

cuC%+/å,Ü8.#%C%2:/oKC=5,#=/>C%&)C%2:/B52+.$'&)&)C/"C

v, wUKN0$'@.+

V2∩NGv(w) = ∅ ⇐⇒ V2∩NG(w) =

V2 \NG(v)U

tA-v2[]C%+9/)(E+ ,Ü8.,#C%2:/go

wc<,N1$'@9+NV[\$'35+.C%@h/>-0$'2 ("@9!%#%!"C%2:/.CL+9B"@

G2"$'B5+4"$'2"2"C-0&I&)!=6"-1/>C&IC2X/

V2 ⊆ NGv(w) ⇐⇒ V2 ∩ NG(w) = V2 \ NG(v)

C/V2 ∩ NG

v(w) = ∅ ⇐⇒V2 ∩ NG(w) = V2 ∩ NG(v)

U ,2"+4/.$'B5/)N1C+)#%,+%cV2 ∩ NG

v(w) = ∅$'B

V2 ⊆ NGv(w)

+9-?C=/+.CB"N1C&IC2:/+9-V2 ∩NG(w) = V2 ∩NG(v)

$'BV2 ∩NG(w) = V2 \NG(v)

UøC/."!%$'@ %&)CL5UTn 2"$'B"+)"$'2"25CLdf,#-1N1C&IC2:/ B"2"C^([email protected]%BAh'CÆN0/>[email protected]Ü/>-0h,C(*$'B5@I&)$'2:/>@9C%@ XB"C^N1df,&)-1N0N1C?"C%+3"-Å6089$'-02X/.+eC%+/K35-76Y(",@9/.-0/>-7h'CU

"$# ÌXZ¾%aÉkÈ&(;(8 :-<=:?>A@CBCBED FHDNL$OQ@CRTSU/@C1+ILiFWV KX1 Y?[:]\+^+DG

DNL]IyOQ@CRE\`:?[aI*@CI@Cb?DQc]>/1`L]I[:?I*@/1mctA$'-7/

FN1ýdf,&)-1N0N1C "C%+ 3"-Å6089$'-02:/>+"C

G = (V,E)U tA$,-0/

v ∈ VC=/F ′ =

V ′ ⊆ V : v 6∈ V ′ C/ V, V \ V ′ ∈ F U,@KN0Cy/."!%$,@7 %&)C5UTnAc

F ′ C%+/N1)df,&)-1N0N1C"C+&)$A"B5N1C%+"CG

v c<C=/4"$'2"# C+9/4B52"C`df,&)-1N1N0C4(E,@/>-0/.-0h,C,U K$'2"# (E@N1CIN1C%&)&)CL'U7"cFC+9/4B52"C`df,&)-1N1N0C43"-76(E,@/>-7/>-0h,C,U

Ã2u(*CB5/@9C%&I,@.XB"C@("N1B5+KM'!2"!%@.,N1C&IC2X/KXB[\B52"C4(*C@.&BA/w/.-1$'2L"C9tAC-1"CN2"C#wE,2"M'C(E,+N1df,&)-1N0N1C?"C%+3"-Å6089$'-02X/.+e[]B"2M'@>,(5"C,UåÈAÉkÉkÈq&W( &( ;/@CI

G = (V,E)KH1Y?[:]\+^+DD]I

W ⊆ Vc BE/[NL

V1, V2DNL]IKX1 OQ@CRTSU/@C1+IFHD

GL]@yD]ILD]KXBED]>D]1+IL]@V1, V2

DNL]I KX1 OQ@CRTSU/@C1+IFHDG

W c

]>/1`L]I[:?I*@/1mc?Ã2(HC%BA/@9C%&I,@.XB"C@DXB"C +9-W = A ∪B

C=/A ∩B = ∅

cEN1$'@9+G

W= G

AB UEWYN+.B )/"$'2"#?"C&I$'2:/[email protected]%@åXB"C(*$'B5@á/.$'B5/

v ∈ VcV1, V2

C+9/B"2 3"-Å6089$'-02:/å"CG+9-*C=/+9C%B"N0C%&)C%2:/+9-

V1, V2C%+/B"2 3"-76789$'-12:/G"C

Gv Uta$'-0/

W1 ⊆ V1

C=/W2 ⊆ V2

/.C%N0+eXB"C_N1C+D,@.=/>C+D"CGC2:/>@.C

V1C/V2

+.$,-1C%2:/GN1C+G@./.C%+GC%2:/>@9CW1

C/W2

cXC/GC%2:/>@9CV1 \W1

C/V2 \W2

UHt5,2"+G(*C@9/>C?"C?M'!%25!%@>N1-0/.!,cv ∈ W1

UÃ2Äh,$'-0/ -1&)&I!"-bÜ/>C%&)C%2:/_XB"C)E,25+G

v cN0C%+y,@.=/>C+C%2:/[email protected]

C/V2

+.$'2:/N0C%+y,@9/>C+C%2:/[email protected] \ v

C/W2

cºC/yC%2:/>@9C(V1 \W1) ∪ v

C/V2 \W2

UK$'2"#V1, V2

C%+9/yB"2Ä3"-Å6089$'-02X/"C

Gv UWY2:h'C@.+.C&IC2:/4+.-

V1, V2C+9/ B"2k35-76089$,-12:/"C

Gv c,N1$,@.+

V1, V2C+9/ B"2k3"-Å6089$'-02:/_"C

Gv

v

= GU

Ã24"-0/<aB5CD"C%BAM'@>(""C%+G = (V,E)

C/H = (V,E ′)

+9$'2:/å!%XB"-7h,N1C%2:/.+ü(E,@üN1(HC%@9&B5/wÜ/>-1$,2"CtAC-1"CNü+%[]-1NüC A-1+9/.C B"2

W ⊆ V/.C%NüXB"C

H+.$,-0/-1+9$'&)$'@.("5CAo

GW UÃ2uh'$'-7/_df,#%-0N1C%&)C%2:/_XB[]-1N+%[\,M'-7/[]B"2"[email protected]%N1/>-0$'2 []!%XB"-7h,N1C%25#%C,UBE,25

GC=/H+9$'2:/K!XB"-0h,N0C%2:/>+(E,@N1 (HC%@9&B5/>/>-0$'2 "C

tAC-1"CNôcN0CN1C&I&)C"U]2"$'B"+"-7/_XB"C Nb df,&)-1N0N1Cy"C%+3"-Å6089$'-02X/.+"CG+9C%@.!M:,N1CoN1 df,&)-1N1N0C 5C%+3"-Å6089$'-02X/.+e"C

HU

ikj _jKtqk tIzoÁxoVtuv zoNk GfV "!%#$'&I(H$'+9-0/>-0$'2 C2#$'B"(HC%+ (HC%B5/=/>@9Cu"!OE25-1CÆ(E@ Nô[\,("("N0-1#Ü/>-1$,2ý+9B"#%#C%+.+9-0h,C«"CuN1k"! 6#%$,&I(H$'+.-7/>-0$'2 C%2 #$'B"(HC%++9-1&)("N1CU2"CI"!%#$'&)(*$'+9-0/.-1$'2Ä+.-0&I("N0C (*CB5/B"+.+9-/[email protected]"!OE25-1C)(*$'B5@N1C3"-Å6089$'-02X/%U

,

tA$'-7/V1, V2

B"2Ä3"-76789$'-12:/K2"$'2 />@.-7h,Nü"CGcøC/y+.$'-7/

W1 ⊆ V1

C=/W2 ⊆ V2

/.C%N0+_aBº[\-0N ,-0/B"2"C89$'-12:/>[email protected]%2:/>@9CW1

C/W2

cHB"2"C?89$'-02:/>B"@9C C2X/[email protected] ⊆ W1

C/V2 ⊆ W2

c*C=/XB[]-1N2[^,-7/K(E+[\,B5/[email protected]%+K,@.=/>C+C2X/[email protected]/V2UH?2"CËÌaÍ'*É«Ç*Î%Á°ÁH rb%HVH Î%VÉkÇÂÁÈ "C

G(E,@?N1C3"-Å6089$'-02X/

V1, V2C+9/G1, G2

c:$ºca(H$'B"@i ∈ 1, 2

cGi

C%+9/GN1C?M'@.,(""Ch'C%#(*$,B"@GC2"+.C&3"N0C"C?+.$,&I&)C/.+Vi ∪ v

c"C=/(*$'B5@?C2"+.C&3"N0Cyº[Q,@9/.C%+ (E ∩ P2(Vi)

)∪v, w : w ∈ Wi

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ikjöi _jKtqk tIzoÁxoVtuv obq%sºtuowvyx "!#%$'&($)&(*+-,/.)01,32547684,/9øC/."!%$,@7 %&)C"U7yC=/?N1C/>5!%$'@ %&)C4'U0,y25$'B"+"$'2"[email protected]#/.C%&)C%2:/i

: Z¾ÜHÂfÂfÀ"f¾°Èi&W(<;;(;/@CIGKX1Y?[7:]\+^+DQc>=]BD@? @EL]IDyKH1JKX1+@BAQK+D:?[=OQ[D

T;B7V CEDFDHGJIKGJIMLONOPRQTS-PRUWVXVYPRZ

F[V\]PRVZ^XyFHDGWID]B_AQK V @CB8`$:?@CIOQ@ SUDba]I@/1D]1+I[NDBEDNL<UD]KH@CBCBEDNL-FHD

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13/D]K`FLg@C1+ID][a13DNLAL/@D]1+I_]I*@BAQK+D]INLJIKLdc5LeZ#LeDHLUq/KfSgDOGeQhVYGeDeU?c#i/KX[gI/KXI OQ@CRTSU/@C1+IV1, V2

FHDG@CBD@? @EL]ID KX1q13/D]K`F

αFHDT

D]II ⊆ 1, . . . , d(α)

ID]BLjAQKDV1, V2 =

⋃i∈I C

iα,⋃

i6∈I Ciαc

5

D\mB K L \+/KH[_I/KXID :?[]IDeFHDTC1

e , C2e

DNL]I KH1qOQ@CRTSU/@C1+I FHDGyD]I;\+/KH[_I/KHI13/D]K`F

αFHDTD]I

∅ ( I ( 1, . . . , d(α)⋃

i∈I Ciα,⋃

i6∈I Ciα

DNL]IKH1 OQ@CRTSU/@C1+IFHDGc

OEM,B"@.C_"U]"U7"$'2"2"CyC%2C=AC&I("N0C_B"2M'@>,(5"Cymh,C%#y+.$'2L,@93"@.C_"C"!#%$'&)(H$'+.-7/>-1$,2 3"-76789$'-12:/U (0$^2 &(** 9#*#%*g2åÈAÉkÉkÈ0&( ;( ;/@CI

G = (V,E)KX1Y?[7:]\+^+D9FHYH]13][-\+/KH[AB : FHbaN/>\+/L]@CI@/1 OQ@CRTSU/@C1+I a+V DNL]I

F?@C[D9ID]B#AQKDgI/KXIDNLgBEDNLOQ@CRE\`:?[aI@CI*@/1`LV1, V2

FHDV:?b?Dba

|V1| ≥ 1D]I|V2| ≥ 1

L/1+I-FHDNLOQ@CRSU/@C1+ICLFHD

Gc+BE/[L

GDNL]I B7V KX1+@/1PF?@ELSU/@C1+ID-FHD FHD]K"? Y?[7:]\+^+DNL aN/>\mBED]ICL]b?D]1+IKD]BCBED]>D]1+Ib@FHDNL

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DNL]I KH1PY?[7:]\+^+DiOQ@\`:?[]I@ aN/>\mBED]Ic]>/1`L]I[:?I*@/1mctA$'-7/

G = (V,E)B"2 M'@.,(""C?h'C#KN1y("@9$'("@9-1!/.!"$'2"2"!cXC/

v ∈ VUMKN1$,@.+

Gv C%+/B"2M'@>(""C_/>CNøaB5Cy(H$'B"@D/.$'B5/

V ′ ⊆ V \ v25$'2ha-0"C,c

V ′ C%+/KB"2&)$A"B5N1C"C Gv UG

v C%+9/"$'25#B"2`M,@>,("5C"!M'!%25!%@.!K"CKN1y"!#%$'&)(H$'+.-7/>-1$,2I&)$A5B"Nb,[email protected]$'2"#G

v C+9/B"2`M'@.,(""C+>,2"+,@.=/>C+%ca$'BG

v C%+9/B"2^M'@.,(""C #%$'&)("N0C/UK,2"+N0C ("@9C%&)-1C@?#%,+%cGC%+/B"2ÆM,@>,("5C 3"-0(E,@/>-#%$'&)("N0C/cZE2"+KN0C"CBAA-E %&)C,c

GC%+/KNV[\B"25-1$'2 "-0+Á89$'-02:/>C"C"CBAM'@.,(""C+?#$'&)("N1C=/>+%U

(0$^2 +##+-6*#"4<2 684$^2$,B5/ü#$'&I&)Cá(*$'B5@N1"!#%$'&)(*$,+.-0/.-1$'2y&)$A"B5Nb,[email protected]/<Nb"!%#$'&)(*$'+9-0/.-1$'2yC%2#$'B"(HC%+c,B"2M,@>,("5C#@>,#=/>!%@9-1+/>-1XB"C (HC%B5/^,B"+9+.-y/[email protected]Ä"!OE25-y(H$'B"@#wE,XB"C«2"$aC%B"½"C«NV[[email protected]"@9C,U?J^,N0"C%[email protected]%B5+.C%&)C%2:/%c#%$,&I&)C_(*$,B"@DNb"!#%$'&)(H$'+.-7/>-1$,2 +.-1&)("N0C,c525$'B"+2[\mh,$'2"+?(E,+"[email protected]%#=/>C&IC2X/?N0C%B"@9+DB"2"-1#-0/.!,UtA$'-7/

G = (V,E)B52½M'@.,(""Cc

T+9$'2,@[email protected]"C "!#%$'&)(*$,+.-0/.-1$'2 3"-76789$'-12:/IC/

αB"22"$aC%B"-12:/.C%@.25Cy5C

TU $'B"@D/>$,B5/

i ∈ 1, . . . , d(α)c"$'2#w"$,-1+.-7/DB"2+.$'&)&)C/

vi ∈ Ciα

UZÃ2L,("(HC%N0N1C_,N0$'@.+N1C_zZ¾°À"Ç # ÈÆÍ'À"¾°ÀEÍ,mÌX¾ÜVÎm;pÈ)"Cαc"2"$/>!

BJC(α)c"N1C_M'@.,(""C

G[v1, . . . , vd(α)]U

åÈAÉkÉkÈ &W( ( @αDNL]IKX1s13/D]K!F "!$#&%'!( :?BE/[L

BJC(α)LD][:KX1ZY?[:]\+^+Di\m[D]>A@D][_\`:?[B :

FHbaN/>\+/L]@CI@/1OQ@CRTSU/@C1+Ic!@αDNL]I KX113/D]K!F*),+.-/+.01+(1+ :?BE/[L

BJC(α)LD][:iKH1AY?[:]\+^+DFHYH]13][

\+/KH[iB :FHbaN/>\+/L]@CI*@/1 OQ@CRTSU/@C1+Ic]>/1`L]I[:?I*@/1mctA-

αC%+/B52«2"$aC%B532,465,487946:,4"cN1$'@9+_(E,@"!O"2"-0/.-1$'2Æ/.$'B5/>C+N1C%+3"-Å6Y(E@9/>-7/>-0$'2"+"C

C1α, . . . C

d(α)α

-12"5B"-1+9C%2:/"C+á35-76089$,-12:/>+"CGUMK$'2"#e/.$'B5/.C%+N1C+ü3"-Å6Y(E,@/>-7/>-1$,2"+5C

V (BJC(α))-125"B"-1+9C%2:/"C+3"-Å6089$'-02:/>+e"CBJC(α)

c"C=/BJC(α)

C%+9/"!M'!%2"!@.!UtA$'-7/KB"2L#%$'B5("N1C

(A,B)/>CNXB"C

A,BC%+/B"25C3"-Å6Y(E,@/>-7/>-1$,2 "C

VUÃ2L2"$,/>Cy(E,@

∼(A,B)

[email protected]/.-1$'2)+.B5@A/.C%N0N1C?aB5C

u ∼(A,B) v+.-N(u)∩B = N(v)∩B

$'B`+9-N(u)∩B,N(v)∩B

C%+/B"2"C(E,@/>-0/.-1$'2 "CBU<;$,/.$'2"+GaB5C

∼(A,B)

C%+/eB"[email protected]/.-1$'2 []!%XB"-7h,N1C2"#%CcAC=/eXB"CA,B

C%+/eB"23"-Å6089$'-02X/+.-øC=/+9C%B"N0C%&)C%2:/?+.-∼(A,B)

(*$'+9+7 "C B52"C+.C%B5N1C#%N1,+.+9Cyº[\!aB5-0h,N1C2"#%CUtA$'-7/

αB"2u2"$aC%B">=?:,@BA CD@E:ZU;tAB"("(H$'+.$,2"+XB"C

BJC(α)2[]C%+/_(E,+("@.C&I-0C%@?(E,@@>,("(H$'@/ o)Nb"!#%$'&)(*$,+.-0/.-1$'2 3"-76789$'-12:/%cÜC=/XB"C

A,BC%+/åB"243"-Å6089$'-02X/2"$'2/[email protected],N

BJC(α) = G[v1, . . . , vk]U

tA$'-7/V1 =

⋃vi∈A C

iα

UKÃ2&I$,2X/[email protected] XB"CV1, B

C+9/LB"2 3"-Å6089$'-02:/ "CG[V1 ∪ B]

U $'B"@/>$,B5/i ∈ 1, . . . , k

/>C%NXB"Cvi ∈ A

C=/)(*$'B5@ /.$'B5/u ∈ C i

α

cu ∼(V1 ,B) vi

c#@Ci

α, V \ CiαC+9/B"2«3"-76789$'-12:/"C

GUWKCI("N1B5+%c(*$,B"@y/.$'B5/

vi, vi′ ∈ Acvi ∼(V1,B) vi′

#,@A,B

C+9/B"2k3"-Å6089$'-02:/"CG[A ∪ B]

UKN0$'@.+∼(V1,B)

B52"C#%Nb+.+.C[]!%XB"-7h,N1C2"#%CB"2"-0aB5CC=/V1, B

C%+9/)B"2Ï3"-Å6089$'-02X/

,

a

d

c e

f

g

i

h

j

b

f

w

v

a

x

d

c e

b

v

g hw

i j

x

g

b

c d

e

j

i

fa

h

5 ]¡Z£5¤E£¦4§ 2`M,@>,("5CKC/+.$'[email protected]"@9CK"C"!%#$'&I(H$'+9-0/>-0$'2)3"-76789$'-12:/UXøC+e2"$aC%B""+G-12:/>[email protected]"C+Dh'C#_B"2M'@.,(""C#@>,#=/>!%@9-1+/>-1XB"C_(H$'+9+.!%",2:/5+9$'&I&)C/.+D+.$'2:/ =,:,@DA CD@6:AcHN1C+,B5/[email protected]%+?+.$,2X/2,465,487946:,4AU

5 " @u

"CG[V1 ∪ B]

U-2«B5/.-1N0-1+>2X/KN1C&)%&)C ,@9M'B"&)C%2:/cN0C3"-Å6089$'-02X/B, V1

"CG[V1 ∪ B]

(*CB5//[email protected]!/.C%2"5B'oÆB52 3"-76789$'-12:/V2, V1

5CGc<h'C%#

V2 =⋃

vi∈B Ciα

UV1, V2

2[\C+9/ (E+E,2"+4NV[[email protected]"@9C"C`5!%#%$,&I(H$'+.-7/>-0$'2k"CGC=/42"$'B5+4h'$'2"+4B52"C`#%$,2X/.@>,5-1#/.-1$'2ºUW-12"+9-

BJC(α)C%+/ ("@.C&I-0C%@y(E,@@>("(*$,@9/o4Nb4"!#%$'&)(*$,+.-0/.-1$'2 3"-76789$'-12:/U

øC%+_M'@.,(""C+#,@.,#/.!%@9-1+9/.-1XB"C+K"C+_2"$ACB""+#$'&)("N1C=/>+K5CNô[\,@[email protected]"!%#$'&)(*$'+9-0/.-1$'23"-76789$'-12:/+.$,2X/á"$,2"#"C%+áM'@>(""C%+å3"-0(E,@/>-1+<#%$'&)("N1C=/>+c'$'B "C%+áB"2"-0$'2 "-1+r89$'-12:/.C%+"C"CBA M'@.,(""C+G#$'&I(5N1C/.+%Uikj _jKtqk tIzoÁxoVtuv |_z |Iyq* X_z

tA-G1, G2

C/G′

1, G′2+9$'2:/LN0C%+ "CBA"!#%$'&)(*$,+.-0/.-1$'25+`+.-0&I("N0C%+`5-*!%@.C2:/>C%+"C

G(",@N1C 35-76089$,-12:/

V1, V2c</.C%N0N1C%+4XB"C

V (G1) = V (G′1) = V1 ∪ v

cü$vC+9/IN1C`&I,@9aB5C%B"@cå,N1$'@9+

G1

C=/G′

1

+.$'2:/y!%XB"-7h,N1C2X/.+(E,@_Nb (HC%@9&B5/wÜ/>-1$,2u"CtAC%-0"C%NVcC/G2

C=/G′

2

!%M:N1C%&)C%2:/%U-2ÄCºC=/G′

1 = G1v C/

G′2 = G2

v UaC#w"$'-Å)"CKN1y"!#%$'&)(H$'+.-7/>-1$,2I+9-1&)("N1CD2[\"$,2"#(",+[]-1&)(H$'@9/>,2"#%C+.B5@GN0C%+á"!%#$'&I(H$'+9-0/>-0$'2"++9-1&)("N1C+á+9B"-0h,2:/.C%+%UAy5!%#%$,&I(H$'+.-7/>-0$'2 +.-1&)("N0C?C+9/G"$'2"#?B"2"-0XB"C$>/QF?KHBE/Nb(HC%@9&B5/>/>-0$'2 "C tAC%-0"C%NVU2ËÈHpº7zZ¾°À"Ç # ÈC%+9/eB"2 #%$'B5("N1C(V,∆)

/.C%NºXB"C∆C+9/DB"25CKdf,&)-1N1N0Ce"C+9$'B"+ö6YC%25+.C%&3"N1C+e"C/w-1N1N0C

3"C

Vc5C=/D/>$'BA/+.$'B"+ö6YC2"+.C&3"N1C_5C/w,-0N1N1C

4"C

V-02"#%N0B5/DB"22"$'&3"@.C(",-1@eº[\C2"+.C&3"N1C+D"C

∆UtAC-1"CNPtaC%-Cv5xÜTø&)$'2:/>@9CXB[\-0NK B52"C3"-¢89C#/>-0$'2`C2X/[email protected]+e"CBAX6YM'@.,(""C+?+9B"@DNô[]C%25+.C%&3"N1C

VC=/N1C+#%N1,+.+9C%+º[\!aB5-0h,N1C2"#%C+_(E,@(*C@.&B5/>/>-0$'2^5CgtAC%-0"C%N"C%+_M,@>,("5C%+[]C%25+.C%&3"N1C+_"C+.$,&I&)C/.+

VUüÃ_2 5-1@>L"$'2"# XB[]B"2"C 3"-Å6Y(E@9/>-7/>-0$'2

V1, V2"C

VC%+9/ B52Ïbr%Hf* []B"2 "CBAX6YM'@.,(""C +9-C%N0N1C C%+9/yB"2 3"-Å6089$'-02:/K[]B"2 M'@.,(""C4"C4N1`#%N1,+.+9C4[\!XB"-0h,N0C%2"#C4#%$'@[email protected]%+9(*$,2"E,2:/ ,Bu"CBAX6YM'@.,(""CU«"!#%$'&)(H$'+.-7/>-1$,2Ï+9-1&)("N1C 5C

H(E,@)B"2 3"-Å6089$'-02:/

V1, V2+.C@>kN1 5!%#%$,&I(H$'+.-7/>-0$'2"C

HC%2

H1

C/H2

c/>CN1N1C)XB"CHi

ci ∈ 1, 2

cC%+/ N1C 5C%BAX6YM,@>,(5"C`#%$,@[email protected]+.(H$'2"E2X/goGi

cG1, G2

C+9/B"2"C "!%#$'&I(H$'+9-0/>-0$'2k+9-1&)("N1C)"CG(E@N1CI3"-76789$'-12:/

V1, V2cC=/

GC+9/ B"2 M,@>,("5C "C`N1^#%N1,+.+9C[]!%XB"-7h,N1C%25#%Cå#%$'@[email protected]+.(H$'2"E,2:/ o

HU[\,("@7 +N0C%+ø$'35+.C%@h/>-0$'2"+("@.!#%!"C%2:/>C+%cÜ#%C=/./>Cá"!#%$'&)(*$,+.-0/.-1$'2+.-0&I(5N1CC+9/?B"2"-0aB5C,U

tA$'-7/TN1Kdf,&)-1N1N0Cå"C%+á"C%Baa6gM'@>(""C%+c:C/

VNbKdÁ$,2"#/.-1$'24XB"-o_B"2 "CBAX6YM'@.,(""C

H,+.+9$A#-1CD+9$'2C%25+.C%&3"N1Cá"Cá+.$'&)&IC=/>+UtA$'-0/

→[email protected]/.-1$'2_XB"-HoB"25C%BAX6YM,@>,("5C

H,+9+.$a#%-1CåB"25CG"!#%$'&)(H$'+.-7/>-1$,2+.-0&I(5N1C

H1, H2(E@?B52L3"-Å6089$'-02:/

V1, V25C

HU

åÈAÉkÉkÈ &(;((T , V,→)

DNL]I KH1 a7:MF?[NDiFHD$FHbaN/>\+/L]@CI*@/1 D]IDNL]IL `>]I*[a@BAQK+DAD]II[:?1`L]@CI*@ <=c

]>/1`L]I[:?I*@/1mcøC%+å("@.$,("@.-0!/>!+VF<C=/VF<+.$'2:/á@.C+.(HC%#/.!%C+(E,@å"!=OE2"-7/>-1$,24"CNb"!#%$'&)(*$,+.-76/>-0$'24+.-0&I(5N1C,UVF<S=cNb("@9$'("@.-0!/.!D"CD+@a&)!/>@9-1CDC=/åNb("@9$'("@.-0!/.!e"Ce/>@.,2"+9-0/>-7ha-0/>!G("@.$mha-1C2"2"C2X/G-1&)&)!=6"-1/>C&IC2X/K"BÆdf,-7/KXB"C4N1Idf,&)-1N1N0C_"C%+K3"-76789$'-12:/>+?"B "CBAX6YM'@.,(""CC+9/B"25Cdf,&)-1N1N0C_3"-76g(E,@/>-0/.-0h,C,U25OE2Nb4("@9$'("@.-0!/.!VFn(5@.$mha-1C%2:/?"Bdf,-0/eaB5CyN14"!%#$'&I(H$'+9-0/>-0$'2`+9-1&)("N1CC+9/?B"2"-0XB"C,UÃ25-1+.(H$'+9C_"$'2"#_[]B"2"C"!#%$'&)(H$'+.-7/>-1$,2`"C%+D5C%BAX6YM,@>,("5C%+%U $'B"@#%C=/./.C_"!%#$'&I(H$'+9-0/>-0$'2caN1C+"CBAX6YM'@.,(""C+#%,@>,#=/>[email protected]+/>-0aB5C%++.$'2:/?B"2"-0XB"C%+U 9ßmÖ×"! Ý=Õ°Ô=×ëÝ Óëæ$#%'&)(+*),-/.0mð

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ikj fVItZoÁxe k _z | | _jtk t zDoÁxoVt vFZ,3"-1C2I5C_JL$'2:/.M'$'N0O"C%@å"$'2"25CKB"2 ,N1M'$,@.-0/."&)CeC%2

O(n2m2)(*$,B"@á/.@.$'BAh'C%@cA+[\-0NEC=A-1+/>CcAB"2I3"-7689$'-02X/25$'2y/[email protected],N'E2"+B52 M'@.,(""CUmWYNX"$'2"2"CáC%25+.B"-7/>CåB"2N1M'$'@9-0/."&ICC%2

O(n3m2)(H$'B"@#,N0#%B"N0C%@øN1"!#%$'&)(*$,+.-0/.-1$'2 3"-76789$'-12:/[]B"2 M'@>,(5"C,c'C24B5/>-0N1-0+>,2:/Nb/.C%#w"2"-0XB"CB5/>-0N1-1+9!%CG(E,@ lB"252"-12"M,E,&(*$'B5@+.$,2 ,N1M'$,@.-0/."&)Cy5C"!%#$'&)(*$'+9-0/.-1$'2LC%2u#%$,B"(*C+A*$,2u"!%#$'&I(H$'+9C/w,2:/_XB"C4#C%N1 C%+/_(*$,+.+.-03"N1C N1C+M'@.,(""C+<(E,@Nb"!%#$'&)(*$'+9-0/.-1$'2y3"-76789$'-12:/+9-1&)("N1CcÜ(5B"-1+$'2dÁB"+9-1$'252"C,cm/w2X/<XB"Ce(H$'+.+9-13"N0C,c°N1C+2"$aC%B5"+"!M'!%2"!@.!+DaB5-º(HC%B5h,C%2://[email protected]ÁB"+9-1$'252"!%+U`KBIOE2E,NVcANV[Q,@93"@.C$'35/.C%2XB C%+/DNô[\,@[email protected]?"C"!#%$'&)(*$,+.-0/.-1$'23"-Å6089$'-02X/%U

2)df,-7/N1CD#%,N1#B"N""CN1"!%#$'&I(H$'+9-0/>-0$'243"-Å6089$'-02X/<(*CB5/á+.CDdf,[email protected])/.C%&)("+åN1-125!,[email protected],cÜC%2)B5/.-1N1-0+>,2:/N1CI/>"!$'@7 &IC"UnAUta-D$'2("@.C2"ýB52Ï+9$'&)&IC=/v"C 5C%M'@9!&)-125-1&B"&c,N1$,@.+N0C`M'@.,(""C

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v c5(H$'B"@D$,35/>C2"-1@eC2 />C%&)("+DN0-12"!%,-1@9CNV[Q,@93"@.CK"C"!#%$'&)(*$,+.-0/.-1$'2 35-76089$,-12:/e"CGU

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GKH1PY?[:]\+^+DQc8DNL FHD]K"?faN/1mF?@CI*@/1`L L]KH@Cb:?1+IDNL L/1+IYAQKH@Cb:?BED]1+IDNL

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T ′ − αC/K"C+.$,2Æ#%$'&)("N0!%&)C%2:/w,[email protected]%+/_B"2Æ35-76089$,-12:/cEC=/_2[]C%+9/(E,+/[email protected],N1CUmtA-125$'2cZ-1NC=A-1+/>C B"2"C#%$,&I(H$'+>2X/.C

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u H C5 s /

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8∀v ∈ V

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|V | ≤ 3U`tA-12"$,2cX-1NHC=A-1+/>CKB52 2"$ACB"

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u ∈ V (G)c<C=/

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v ∈ V (G) (E,@-12"5B"#/.-1$'2c

Gv C%+/?B"2L#$'M'@.,(""CU3ta-

u 6= vcwC%+/?B"2Lha@>-89B"&)C,B 5C

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u = vC/wC%+9/B52 dfBA)89B5&IC%,Bu$'B B"2ha@>,-Z2X/.-76789B"&IC%,B "C

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(2) ⇐⇒ (4)+tA$'-7/

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v ∈ V (H)UXKN0$'@.+

Hv C%+/B"2

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HC%+/?B"2C5

cEB"2/>,B"@9C,BcEB52M,C%& $'BB"2#%$6gM'C%&UtA$'-7/

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v ∈ VcG

v C%+/?B"2#%$'M'@.,(""CUtA$'-7/HB"2+.$,B"+6gM'@>,(5"C-125"B"-0/5C

GcZC/?+.$,-0/

v ∈ V (H)U+KN0$'@.+

Hv

= Gv[V (H) \ v]

2[]C%+9/(E,+?B"2P4

cH5$'2"#H2[]C%+/(E,+ B"2

C5

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C5

c/wB"@.C%,Bc:M'C& C/á#$6YM'C& -02""B"-7/c,C=/á+.$'-7/v ∈ V

UHtA$'-7/H ′ B"2)+.$'B"+ö6YM'@.,(""C"C Gv U

G[V (H)∪v]2[]C%+/_(E,+B"2C5

cHB"2^/wB"@.C%,BcHB"2uM'C%& $'B^B"2Æ#%$Ü6YM'C&LcZ"$'25#H ′ = G[V (H) ∪ v]

v 2º[\C+9/_(E+B"2P4

U #%Nb+.+.C^"C%+ M'@.,(""C+ +>,25+

C5

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v cX$vC%+9/eB"2 +9$'&I&)C/G"C"C%M'@9!K&)-12"-0&B"&U

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l$'&)&IC_(H$'B"@Nb4"!#%$'&)(H$'+.-7/>-1$,2 &I$a"B"N1,-1@9CC/?Nb4"!#%$'&)(H$'+.-7/>-1$,2 C%2#%$'B"(HC%+c*N1Nb,@9M'C%B5@D"C#%N0-1XB"C[]B"2 M'@.,(""CLC%+9/`3*$'@92"!%CL+.-?C/ +.C%B5N1C%&)C%2:/ +.-?Nb«N1,@.M,C%B"@ "CL#%N1-0XB"CL5CÆ#wE,XB"CÆM,@>,("5C#@>,#=/>!%@9-1+/>-1XB"C_5CyN14"!%#$'&I(H$'+9-0/>-0$'2`3"-Å6089$'-02:/DC%+/3H$'@92"!%CU"$# ÌXZ¾%aÉkÈq&()H (-;/@CI

GKH1 Y?[:]\+^+DQcBE/[L

cwd(G) ≤ 8 × k/

kDNL]I B :GB :?[*YHD]KX[FHD a]B @BAQK+D

>g: ? @C>AKH> FHDNL-Y?[:]\+^+DNLja7:?[:>a]I][a@EL]I@BAQKDNLiFHDNLi13/D]K!FL @C1+ID][a13DNL FHDiB :JFHbaN/>\+/L]@CI@/1OQ@CRTSU/@C1+IFHDGcWYN2"C+.C%@.«(E+)5!%&)$'2:/>@.!-1&)&)!%"-1/>C&IC2X/%UWYN+[Q,M,-0/ []B"2 #$'@.$,[email protected]"C%+ />5!%$'@ %&)C%+x5U0,C/-x5Uux4"B#wE,("-7/>@.C x XB"-ø(E,@9N1C@>C%2:/[email protected] ,B5/>@9C"[email protected]'CB"@e&)$A"B5Nb,[email protected],U,2"+N0C#,+(E@9/>-0#%B"N0-1C@"C+DM'@>,(5"C%++>2"+

C5

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a, b, c, d, eB"2)/w,B"@9C,B"C

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a, b, c, d+.$,-0/?B"2

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M ′V

Nô[]C%2"+9C%&3"N0Cv ∈ V \ a, b, c, d, e : NG(v) ∩ a, b, c, d, e = V ′

UKN0$'@.+Mb

cMa,b

cMc

cMb,c

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cMc,d

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C=/Mb,c,d,e

dÁ$'@9&IC2:/B"2"C (E,@/>-0/.-1$'2^"CV \ a, b, c, d, e

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c5M'C& C=/D#%$Ü6YM'C& -02""B"-7/UH,25+D#%CK/>,3"N1C%,Bc +9-1M'25-0OECaB5CN1C+"CBA#%N1,+.+9C%+?2"C_+.$,2X/?(E+Ü8.,#%C2:/>C%+cy+.-0M'2"-7OEC_XB[\CN1N0C%+D+.$,2X/,°8.,#%C2X/.C%+?C/4+9-1M'25-0OECXB[]C%N0N1C+"$'-7h'C2X//[email protected] dÁ$'-1+y,°8.,#%C2X/.C%+C=/2"$,2 ,Ü8.#%C%2:/.C%+%c#,[]C%+/[email protected])XB"C`NV[\B"25CI[]C%N0N1C "$'-7/y/>@9Cha-1"CU

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a Mb

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+.$'2:/`/[email protected],BA*cåC/ "$,2"#Mb = ∅

C=/Ma,b = ∅

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cMc,d = ∅

cMb,c = ∅

cMb,c,e = ∅

cMa,c,e = ∅

cMa,b,c,e = ∅

cMb,d,e = ∅

C=/Mb,c,d,e = ∅

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+.$'2:/"C+D&I$a"B"N0C%+e5CGc",N1$'@9+

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c|Ma,b,d| ≤ 1

C=/|Ma,c,d| ≤ 1

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5U

$'B5@/.$'B5/i ∈ 0, 1, 2, 3, 4

cA+9$'-0/_§Xi = v ∈ V \ C : N(v) ∩ C = vi−2, vi+2

c§Yi = v ∈ V \ C : N(v) ∩ C = vi−2, vi, vi+2

c§Zi = v ∈ V \ C : N(v) ∩ C = vi−1, vi+1

c§Ti = v ∈ V \ C : N(v) ∩ C = vi−1, vi, vi+1

UÃ2^V =

⋃4i=0(Xi ∪ Yi ∪ Zi ∪ Ti ∪ vi)

UøC/>,3"N0C,B "CN1OEM'B"@9C_"U 4&)$'2:/>@9C_N1C%+Ü8.,#%C2"#%C+C2:/>@.CN0C%+DC%25+.C%&3"N1C+?5!OE2"-0+("@.!#%!%5C%&46&)C%2:/UEN1N0C%+e(5@.$mha-1C%252"C%2:/"-1@9C%#/.C%&)C%2:/?"Bdf,-0/DXB"CGC+9/?+>2"+M'C%& C=/?#%$6gM'C%& -02""B"-7/UÃ2Ï(HC%BA/)[email protected] aB5C%N1XB"C+I$,3"+.C@9h/>-0$'2"+4+9-1&)("N1C+%UüøC+)XBEÜ/>@.C ([email protected]%&)-E @.C+4$'3"+.C@9h/.-1$'2"+4+9$'2:/B"2"C4@9C/.@>,2"+9#%@9-1(5/.-1$'2u"C+("@.!#%!%5C%2:/>C+,Ü8.#%C%25#%C%+UøC%+,B5/[email protected]+y#$'2"#C%@.25C%2://>@.$,-1+C2"+.C&3"N1C+$'B("N0B"+%U

Á/% tA-Xi 6= ∅

cZ,N0$'@.+Xi+1 = Xi−1 = ∅

UÁ/Nn tA-

Yi 6= ∅cZ,N0$'@.+

Yi+2 = Yi−2 = ∅U

Á/>S tA-Xi = Yi = ∅

cZ,N0$'@.+Zi ∪ Ti ∪ vi

C+9/?B"2&I$a"B"N0C"CGU

Á/. t*[]-1NºC A-1+9/.C_B"2i/>CNXB"C_(*$,B"@e/>$'BA/

j 6= icXj = Yj = ∅

c",N0$'@.+Xi ∪ Yi ∪ Zi ∪ Ti ∪ vi

C+9/B"23"-76789$'-12:/D"CGU

u H s

Zi+

1

Ti+

1

Xi+

1

Yi+

1

Zi+

2

Ti+

2

Xi+

2

Yi+

2

Zi+

3

Ti+

3

Xi+

3

Yi+

3

Zi+

4

Ti+

4

Xi+

4

Yi+

4

Zi

Ti

Xi

Yi

5 ]¡Z£º¤E£?¦u§ DÜ8.,#C%2"#CC%2:/>@9CN1C+K"-+º[email protected]/.C%+#Nb,+9+.C+K"CN1I(E@9/>-7/>-0$'2E,2"+N0C #%,+Kº[\B"2uM,@>,("5C+>2"+M'C%& C=/?#%$6gM'C%& (H$'+9+.!%",2:/B52C5

UÁ- GtA-

Xi 6= ∅C=/Xi+2 6= ∅

c,N0$'@.+Xi

C%+/)2"$'2Ï,Ü8.,#C%2:/oZi

cTi

C=/Yi

cüC/Xi+2

C%+/I25$'2,Ü8.#%C%2:/-oZi+2

cTi+2

C=/Ti+2

Uf-0- GtA-

Yi 6= ∅C=/Yi+1 6= ∅

c5N1$'@9+Yi

C%+/?,Ü8.,#C%2:/oZi

cTi

C=/Xi

cAC/Yi+1

C%+/,Ü8.#%C%2:/oZi+1

cTi+1

C=/Xi+1

UÁ-1-1- JtA-

Xi 6= ∅C/Xi+2 6= ∅

cE,N0$'@.+Yi+1 = ∅

Uf-0h tA-

Yi 6= ∅C/Yi+1 6= ∅

cZ,N1$,@.+Xi−2 = ∅

Urh PtA-

Xi 6= ∅,N1$,@.+

Yi+1

C+9/,Ü8.#%C%2:/-oZi+1

C=/Ti+1

cEC=/Yi−1

C+9/,Ü8.,#C%2:/-oZi−1

C/Ti−1

UÁha- tA-

Yi 6= ∅,N1$,@.+

Xi+2

C+9/2"$'2ÆÜ8.,#%C2:/ oZi+2

C=/Ti+2

cEC/Yi−2

C+9/2"$'2^,Ü8.,#C%2:/ oZi−2C/

Ti−2

UÁha-0- GtA-

Yi 6= ∅,N1$,@.+

Xi+1C%+/25$'2,Ü8.,#C%2:/-o

Zi+1C/Ti+1

c"C=/Xi−1

C+9/?2"$'2,°8.,#%C2X/-oZi−1C/

Ti−1U

rha-1-1- JtA-Xi 6= ∅

,N1$,@.+Yi+2

C+9/,Ü8.#%C%2:/-oZi+2

C=/Ti+2

cEC=/Yi−2

C+9/,Ü8.,#C%2:/-oZi−2

C/Ti−2

Uf-7 tA-

Yi 6= ∅,N1$,@.+

Xi+1C+9/,Ü8.#%C%2:/-o

Yi+1cEC/

Xi−1C%+9/,°8.,#%C2X/-o

Yi−1U

b PtA-Xi 6= ∅

,N1$,@.+Yi+2

C+9/?2"$'2L,Ü8.#%C%2:/-oXi+2

cEC/Yi−2

C%+/25$'2^Ü8.,#%C2:/oXi−2

UB"-1+9aB5C

GC%+9/("@9C%&)-1C%@(E,@@>,(5(*$'@/o4Nb"!#%$'&)(*$,+.-0/.-1$'2 35-76089$,-12:/ca$'2L

rA- tA-Xi = Yi = ∅

cZ,N0$'@.+Zi = Ti = ∅

f"CNô[]$'3"+9C%@h,Ü/>-1$,2/>S=UrA-0- Gt*[]-1N*C=A-1+/>C

i/>CNXB"C(*$,B"@/>$,B5/

j 6= icXj = Yj = ∅

cE,N0$'@.+GC%+/B"2

C5

Á"C_Nô[]$'3"+9C%@h,Ü/>-1$,2/>=UbA-1-1- JtA-

Xi 6= ∅C=/Xi+2 6= ∅

c,N0$'@.+|Xi| = |Xi+2| = 1

cø#%,@_(E,@Nô[]$'3"+9C%@h,Ü/>-1$,2u-ôcXi

C/Xi+2+.$,2X/?"C+&)$a"B"N1C+%U

rA-0h tA-Yi 6= ∅

C=/Yi+1 6= ∅

cE,N0$'@.+|Yi| = |Yi+1| = 1

cE#@?(",@DNô[]$'3"+9C%@9h/.-1$'2-0-ôcYi

C/Yi+1

+9$'2:/"C+&)$A5B"N1C+%UÃ2Ï@.,("(HC%N0N1C XB[]-1N2"C (*CB5/ h'$,-1@4,B("N0B"+XB"C"CBA-125"-1#C%+"-+º[email protected]/.+

iC/j/>C%N0+4XB"C

Xi 6= ∅C=/Xj 6= ∅

cHC/KE,2"+K#%C#%,+j = i + 2

$'Bj = i − 2

U KC&)%&)C,cH-1N2"C(HC%B5/^h,$'-1@K,B("N0B"+XB"C_"CBA -12""-0#%C+e"-*!%@9C%2:/>+iC/j/.C%N0+eXB"C

Yi 6= ∅C/Yj 6= ∅

c5C=/DE,2"+#%C#%,+j = i+ 1

$'Bj = i− 1

UHÃ2!/.B""-1C&I,-12:/.C%2E,2:/DN0C%+"-+º[email protected]%2:/.+?#%,+%Ule,+yg5(*$'B5@/.$'B5/

icXi = ∅

Ule,+y'U09"(H$'B"@D/.$'B5/

jcYj = ∅

U!N1$'@9+GC+9/B"2

C5f$'35+.C%@h/>-0$'2`5-0- U

,

le,+'UnYj 6= ∅

(H$'B"@GB"2`B"2"-0XB"CjUXN1$'@9+

G(H$'+.+ %"C_B52`3"-76789$'-12:/á2"$'2I/>@.-7ha-b,N f$,3"+.C@9h/>-0$'2/>=Ul$'2:/.@>,"-0#/.-1$'2U

le,+ 'U\S Yj 6= ∅

C=/Yj+1 6= ∅

U3N1$'@9+%c*(E,@_Nô[]$'3"+9C%@h,Ü/>-1$,2LA-0hHcYj

cYj+1

+.$'2:/"C+y&)$a"B"N1C+%U$,B"@ />$,B5/

k 6∈ i, i + 1cZk = Tk = ∅

Á$'3"+.C@9h/.-1$'2ýA- =cC=/ (H$'B"@ /.$'B5/k ∈ i, i + 1

cZk ∪ Tk ∪ vk

C%+/B"2 &)$A"B5N1C Á$'3"+9C%@9h/.-1$'2 - =UHK-02"+.-VcGC+9/DN0C+.$,B"+6gM'@>,(5"C_"C

G1-02""B"-7/e(E@

a, b, c, d, e, h, iU

le,+nGXi 6= ∅

(H$'B"@DB"2B"2"-0XB"CiU

le,+_naU0gYj = ∅

(H$'B"@/>$'BA/jUKN0$'@.+

G(H$'+.+ %"CB"2^3"-Å6089$'-02:/2"$'2L/[email protected],N Á$'3"+.C@9h/.-1$'2/> U

l$'2:/>@.,"-0#/>-0$'2Ule,+nAUTnJ

Yj 6= ∅(H$'B"@DB"2B"2"-0XB"C

jU

le,+naUTnAU7i = j

UN1$'@9+G(*$'+9+7 "CB"2 3"-76789$'-12:/2"$,2y/[email protected],N f$,3"+.C@9h/>-0$'2 /.=Ul$,2X/.@>,5-1#/.-1$'2ºU

le,+AnaUTnAUn i ∈ j + 1, j − 1

c(",@ NV[\$,3"+.C@9h/>-0$'2 h*cYj

C%+/4,Ü8.#%C%2:/oZj

C=/Tj

cC=/(E@Nô[]$'3"+9C%@h,Ü/>-1$,2hA-0-ôcXi

C+9/á2"$'2),Ü8.#%C%2:/oZi

C=/Tj

U KN0$'@.+Xi

cYj

c:C=/å(*$'B5@å/.$'B5/kcZk∪Tk∪vk+.$,2X/?"C+&)$a"B"N1C+%UXK-125+.-ôc

GC%+/KN0C+.$,B"+6gM'@>,(5"C 5C

G1

-12""B5-0/(E,@a, b, c, d, e, f, h

Ule,+nAUnAU\SJ

i ∈ j + 2, j − 2U+l$,&I("N0!%&)C%2:/[email protected]?"B #%,+nAUnAUTnaU

GC%+9/N0CK+9$'B"+6gM'@>(""C_"C

G1-125"B"-0/e(E,@a, b, c, d, e, f, i

Ule,+ynAUQS9

Yj 6= ∅C/Yj+1 6= ∅

UMKN0$'@.+c:(E,@åNô[]$'3"+9C%@9h/.-1$'2 -7h*ci 6= j−2

UXtA-i ∈ j, j+1

c'+.,2"+(HC%@9/.C"CM'!%2"!@>,N0-0/.!$'2L,+.+9B"&ICi = j

U ,@N1C+D$'3"+.C@9h/.-1$'2"+-0-ôcXh C/DA-ôcGC%+9/?N0C_M'@>,(5"C

G3U

tA-i ∈ j − 1, j + 2

cA+.,2"+D(HC%@9/.C_"CM'!2"!%@.,N1-7/>!$'2,+.+9B"&)Ci = j − 1

U <,@eNV[\$,3"+.C@9h/>-0$'2 -7hHcXi

C=/Zi ∪Ti ∪vi

+9$'2:/e"C+&)$A"B5N1C%+U <,@áN1C+$'3"+9C%@h,Ü/>-1$,2"+áh)C=/Gha-1-0-ôcYj

cYj+1

cZj ∪Tj ∪vjC/

Zj+1 ∪ Tj+1 ∪ vj+1+.$'2:/I"C%+)&)$A"B5N1C%+UK$,2"#

GC+9/)N1C+9$'B"+6gM'@>(""C"C

G1

-12""B5-0/4(E@a, b, c, d, e, f, h, i

Ule,+S

Xi 6= ∅C=/Xi+2 6= ∅

Ule,+_S5U7G

Yj = ∅(*$,B"@/.$'B5/

jU3l$'&I(5N1!%&)C%2:/>,-1@9C "Bu#%,+'U\S5U

GC%+9/N0C+9$'B"+6gM'@>(""C"C

G1-125"B"-0/e(E,@a, b, c, d, e, f, g

Ule,+eSAUTn9

Yj 6= ∅(H$'B"@áB"2`B"2"-0XB"C

jUXl$'&)("N1!&IC2:/w,[email protected]"B #%,+naUQS5U

GC+9/G+.$'-7/N0C?M,@>,("5C

G2

c+.$,-0/N1C_+9$'B"+ö6YM'@.,(""C"CG1

-02""B"-7/D(E,@a, b, c, d, e, f, g, h

Ule,+áS5U\S

Yj 6= ∅C/Yj+1 6= ∅

U ,@åN1C+ü$'3"+9C%@9h/.-1$'25+-1-1-aC/á-7h*cj = i+3

U <@N1C%+$,3"+.C@9h/>-0$'2"+hC/?ha-ôcXi Xi+2

cYi+3

C=/Yi+4

+9$'2:/"C+K&)$A5B"N1C+%UKC ("N1B"+?(H$'B"@/>$'B5/kcZk ∪ Tk ∪ vk

+.$'2:/"C+&)$A5B"N1C+%U!$'2"#GC%+/?N1C_M'@.,(""C

G1U

"$# ÌXZ¾%aÉkÈg&() ;;( ;/@CIGKH1gY?[:]\+^+D LQ:?1`LYHD]> D]I aN/RYHD]> @C1mF?KH@CI D]I

TL/1 :?[=OQ[D-FHDFHbaN/>\+/R

L]@CI@/1 OQ@CRTSU/@C1+Ic BE/[LBEDNLY?[:]\+^+DNL a7:?[7:>a]I][a@EL]I*@BAQK+DNL_FHDNL 13/D]K`FL "!$# %<!(FHDTL/1+I FHDNL-L/KML]R

Y?[:]\+^+DNL @C1mF?KH@CIL$FHDNL-Y?[:]\+^+DNL FH/1+13NL D]1 Y?KX[DNL mcGD]I mc D]IyFH/1Ka/1+Iy:?K9\mB KML9L/>A>D]ICL?c

WYN_C%+/^#$'2"2XB½5CýPeJ^R,TaB5C«N1 Nb,@9M'C%B"@ 5CÄ#N1-0aB5Cu"C%+M'@>(""C%++.,2"+M'C%& C=/#%$6gM'C%&-125"B"-0/C%+/3*$'@92"!%C_(",@16U5øC/>5!%$'@ %&)C"C_"!#%$'&)(*$,+.-0/.-1$'2I("@.!#%!"C%2:/2"$'B"+e5$'2"2"C_B"25C_("@.CB5h'C,N7/>C%@92E/.-0h'C_C=/("N1B5+?#$'B"@/>C(*$,B"@ )@.&)C%@DXB"Cy#%C=/./>C#Nb,+9+.Cy4B"2"CyNb,@9M'C%B"@"C#%N1-0XB"C_3*$,@.2"!C,U

åÈAÉkÉkÈ &W()< ;(8;:B :?[*YHD]KH[ FHD a]B @BAQK+D FHDNLY?[7:]\+^+DNL_FH/1+13NL$D]1 Y?KX[D mcD]I mc DNL]I >g:SU/[\`:?[6c]>/1`L]I[:?I*@/1mc[\C 5([email protected]%+9+.-0$'2`+.B5-0h,2:/>CC%+/B52"C

66YC 5([email protected]%+9+.-0$'2I(H$'B"@åN0C?M,@>,("5C?5CN1_O"M'B"@.C"Uux5U

t0 = η4,5η4,6η5,6η4,1η5,3η6,2((a)4 ⊕ (b)5 ⊕ (h)6 ⊕ ρ6→1ρ5→3ρ4→2(t′0))

$t′0 = η1,2η1,2η2,3η2,4η3,5η4,5η5,6η4,6((i)1 ⊕ (d)2 ⊕ (c)3 ⊕ (g)4 ⊕ (e)5 ⊕ (f)6).øC%+ C=A("@9C%+.+9-1$'2

t1ct2C/t3+.$'2:/)"C+)C A("@.C+.+.-0$'2"+ (H$'B"@4N1C+4M'@>(""C%+

G1

cG2

C=/G3

"C N1OEM'[email protected]"U 5U

t1 = η4,5η4,6η5,6η4,1η5,2η6,3((a)4 ⊕ (i)5 ⊕ (e)6 ⊕ ρ6→3ρ5→2ρ4→1(t′0))

t2 = η6,3η6,4η6,5((g)6 ⊕ η3,2ρ6→3η6,5((h)6 ⊕ (f)3 ⊕ ρ3→2(tC5)))

t3 = η6,3η6,2η6,1(g)6 ⊕ η4,3η4,2ρ6→4η4,6η6,5((f)4 ⊕ (h)6 ⊕ ρ4→1(tC5))$t′1 = η3,6η3,4η3,5η2,6η2,5η1,5η1,4η1,2((b)1 ⊕ (f)2 ⊕ (d)3 ⊕ (h)4 ⊕ (c)5 ⊕ (g)6)C/

tC5 = η5,1η4,5η3,4η2,3η1,2((a)1 ⊕ (b)2 ⊕ (c)3 ⊕ (d)4 ⊕ (e)5).

øCyN0C%&)&ICK"U0Ux C=/?N1C/>5!%$'@ %&)C_"U0n425$'B"+"$'2"25Cy-0&I&)!%5-b/.C%&)C%2:/ : Z¾ÜHÂfÂfÀ"f¾°È &W()? ( 8 :PB :?[YHD]KH[FHD a]B @BAQK+DFHDNL Y?[:]\+^+DNLLQ:?1`L YHD]> D]I aN/RYHD]> @C1mF?KH@CI DNL]IO]/[a13ND\`:?[

48c

Ã2k$'35/.-1C2X/yB"25CI&)C%-0N1N1CB"@.C43H$'@92"C +.-å$'2k(",+.+9CI"[email protected]#/>C&IC2:/(E@yN1N1,@.M,C%B"@_&)$A5B"Nb,[email protected]@.M'CB"@&)$a"B"[email protected]_5C%+M'@.,(""C+KC2^OEM'B"@9C%+5Ux)C/5U C+9/K3*$'@92"!%C(E,@3U5å2^B5/>-0N1-1+.,2:/?N1C/>"!$6@7 &ICx5U7''c$'2kh,$'-0/ XB"C`NbNb,@9M'C%B5@ &)$A5B"Nb,[email protected]"C+M'@.,(""C++>2"+M'C& C=/#$6YM'C&-12""B5-0/>+yC%+/3H$'@.2"!C(E,@

12c"C/?"$'25#,cE(E,@N0C/>"!$'@7 &IC x5Uux5cEN0C%B"@DN1,@.M'CB"@e"C#N1-1XB"C_C+9/?3H$'@.2"!C(E,@

24U

ikjml t v$j9fwzoVtuvK-02"+.-+%[\,#w! =h'C2"$/>@.C !/.B""C5C#%C=/./.C2"$'BAh'C%N0N1Cy"!%#$'&)(*$'+9-0/.-1$'2UZ)"!%#$'&I(H$'+9-0/>-0$'23"-76789$'-12:/C%+/"$'25#eB"2"Ce5!%#%$,&I(H$'+.-7/>-0$'2 ",2"+üN1KN1-0M'2"!%Cá"CeN1"!#%$'&)(H$'+.-7/>-1$,2 &)$A5B"Nb,[email protected]åC/ü"CeNb?"!%#$'&)(*$6+.-7/>-0$'2 C%2 #$'B"(HC%+U5Ã2 yNV[\B"25-1#%-7/>!e5CNV[Q,@93"@.C@9C%("@9!%+.C2:/w,2:/eN0C%+á3"-76789$'-12:/.+%c,C=/([email protected]%+9XB"CKNV[\B"25-1#%-7/>!D"C+M'@.,(""C+D#,@.,#/.!%@.-0+9/.-1XB"C%+Á$'24Nô[]B"2"-0#%-0/.! >/QF?KHBE/Nb(HC%@.&B5/w/.-1$'2 "CitAC%-0"C%N =U5Ã25-1+.(H$'+9C_!%M:Ü6N1C&IC2:/å[\B52 ,N0M'$'@9-0/>5&ICGN1-125!,[email protected](H$'B"@#%,N1#B"N1C@#%C/9/>C"!#%$'&)(*$,+.-0/.-1$'2ºc,C/á("N0B"+.-0C%B"@9+å#,@.,#/.!%@.-0+>Ü6/>-0$'2"+å"C+GM'@.,(""C+G#$'&)("NE =/>C%&)C%2:/á"!#%$'&)(*$,+>,3"N0C%+U`lC=/./.C"!#%$'&)(H$'+.-7/>-1$,2)(HC%BA/G=/>@9CM'!2"!%@.,N1-0+.!C,BA "C%BAX6gM'@>,(5"C%+cHC=/?E,2"+#%C#%,+$'2$'35/>-0C%2:/?Nô[]B"2"-0#%-0/.!"C+?5C%BAX6YM,@>,("5C%+?#,@.,#/.!%@.-0+9/.-1XB"C%+U2"CXB"C+9/>-0$'2)$'B5h'C@9/.CC+9/á"C+>h'$,-1@+.-E$,2I(HC%B5/åM,!%2"!@>,N0-1+.C@#%C=/./>C"!#%$'&)(H$'+.-7/>-1$,29oº[Q,BA/>@.C++9/[email protected]"#=/>B"@9C%+cH#$'&)&IC_N0C%+DM'@.,(""C+?$,@.-1C2:/>!%+cEN1C+

k6Y+/>@.B5#/>[email protected]%+?$'BN1C+D[A(HC%@9M'@>(""C%+U

2«("("N1-0#/.-1$'2ºcH$,2Ä5$'2"2"C4B"2u/."!%$'@ %&)C4"C4"!#%$'&)(*$,+.-0/.-1$'2L"C%+M'@.,(""C+_+>,25+yM,C%& C=/_#%$Ü6M'C& -12""B5-0/c,#%C?aB5-H25$'B"+á"$'2"2"CB"2"C?("@.CB5h'CK,N0/.C%@92E/>-7h'CDXB"CN1C%B5@áN1,@.M,C%B"@"C?#%N0-1XB"CC%+/3H$'@92"!%CU25Cu,B5/[email protected]^XB"C+9/.-1$'2 +9C%@>-0/I"C^/>@9$'B5h,C%@`[\,B5/[email protected]+`#%N1,+.+9C%+y89$'N0-1&)C%2:/)"!%#$'&)(*$'+.,3"N0C%+ (",@ #C/./.C"!#%$'&)(*$,+.-0/.-1$'2ºcC=/4"C`/.@.$'B5h,C%@º[Q,BA/>@.C+4("@.$,3"NE &IC+2E/.B"@.CN1+XB"-+.C`#$'&I(H$'@/>C%2:/435-1C%2k(E,@N1"!#%$'&)(*$,+.-0/.-1$'2 35-76089$,-12:/U

KC2"$'&3"@9C%BAÏ(5@.$'3"N %&)C%+ $'2:/I!/>!&)$'2:/>@.!+ (*$'N7a2"$'&)-b,BAk+.B5@IN0C%+ #%N1,+.+9C%+ 5CLM'@.,(""C+ "[email protected]'CB"@?,@.3H$'@.C+.#C%2:/>C3H$'@92"!%CU;lK[]C%+/N1Cy#+"Cy/>$'B5+N0C%+K(5@.$'3"N %&)C%+XB"-(*CB5h'C2X/+%[]!%#%@9-1@9CC%2^N1$6M'-0aB5C&I$,2E,"-0aB5C"B +9C%#$'2"$'@[email protected] Wt 5TôU5C+e("@9$'3"NE &IC+ !"$#&%'(*),+ #&%'c-.!/"$#0%,132 "$45.+ .)üc768459&68:.4;)=<>+3"$45%?) 2 45C%24dÁ$'2:/(E,@/>-0C,U'?2"C,B5/[email protected]#Nb,+9+.CD"CD(5@.$'3"N %&)C%+ü5CD(E,@9/.-76/>-0$'2"2"C&IC2:/5CNô[]C%25+.C%&3"N1Cå"C+<+.$,&I&)C/.+C+9/<!%M:,N0C%&)C%2:/(*$,N<a2"$'&)-b,N0C<+9B"@<N0C%+M'@>(""C%+"CáNb,@9M'C%B"@,@93*$'@9C%+9#%C%2:/.C_3*$'@92"!%C P MvÜTVUEÃ2 (HC%B5/2"$/>C%@aB5C_#%C%N1C%+/!%M:N1C%&)C%2:/N1CK#%,+D(H$'B"@eN0CK("@9$'3"NE &IC45 2 " 2 9>@< 45!"$ 1 BAC98+3@<.LP$A 'RÜTVc'(H$'B"@"!#%-15C%@å+.-EB"2 M'@.,(""C?B"25C?N1,@.M,C%B"@@.3H$'@.C+.#%C2:/>C3H$'@.2"!C(",@GB52

kO5A!P$A xÜTHC/G(H$'B"@å"!%#-1"C@G+9-*B52 M'@.,(""CyB52"CKN1,@.M'CB"@å"C?#%N1-0XB"C[QB ("N0B"+

kPED$FR'SÜTôUE?2"ChaB"C_M'!2"!%@.,N1C5C%+DB5/.-1N1-0+>/.-1$'25+G"[email protected]'CB"@e,@93*$,@.C%+9#%C2X/.C(*CB5//[email protected]/>@9$'B5h,!%CE,25+ P$a'S°TôU$,B"+IN0C%+ ("@9$'3"NE &IC+4XB"-(*CB5h'C2X/ +%[]!%#@.-1@9CC%2ÏN1$'M'-0XB"C &I$,2E,"-0aB5C "Bý+9C%#$'2" $,@."@9CLh,C%#XBE,2:/>-7OE#Ü/>-1$,2k+.B5@N1C++.$,&I&)C/.+ (HC%B5h,C%2:/ /[email protected]`@9!%+9$'N1B"+ C%2k/.C%&)("+N0-12"!%,-1@9C4+.B"@N0C%+ M'@>,(5"C%+ "[email protected]'CB"@"C4#N1-1XB"C3*$'@92"!%CPulDJHG?R'R,TVU;$,B"+_N1C+_("@.$'35NE %&)C%+K[\B52"C #%Nb+.+.C45C4("@.$'35NE %&)C%+K"C (E,@6/>-7/>-0$'2"2"C&IC2X/ fB52"C^+9$'B"+6g#%N1,+.+9CÆ"CÆ#C%N1N0CL5$'2"2"!CÆE,2"+ÆP MvT )(HC%B5/ /[email protected]Æ@9!%+.$,N1B"+IC%2 />C&I(5+(H$'N<a2"$'&)-bN<PID leR,SÜTVcEC/ P EDJFR5%Tø5$'2"2"CB"2"CyN1-0+9/.C"C("@.$,3"NE &IC+D(*$'N7a2"$'&)-b,BA*U,2"+D#C#wE,(5-0/>@9C,ca$'2 &)$'2:/>@.C@>XB[]B"2"C_25$'B5h'CN1N0CK#%N1,+.+9C"C(5@.$'3"N %&)C%+G"C(E,@/>-7/>-1$,2"2"C%&)C%2:/"C?Nô[]C%2"+9C%&35N1C?"C%+G+.$'&)&IC=/>+åC%+/e(H$'N<a2"$,&I-1,N1Ce+9B"@áN1C%+áM'@.,(""C+G"CNb,@9M'C%B5@á5C#N1-1XB"C?3H$'@.2"!C,UAÃ2C%2"!"B"[email protected]º[\B"25C,BA/>@.C_#Nb,+9+.C"C("@9$'3"NE &IC+%c"(5N1B"+DM'@.,2""CcEC%+/(H$'N7A25$'&I-1,N1C?+.B5@?N0C%+DM'@.,(""C+"[email protected]'CB"@,@.3H$'@9C%+.#C%2:/>Cy3H$'@.2"!C,U

kjml t fVtXyxoVt v zåvzH

owv|k oÁx+.(HC%NbM'C,c0D_B"@.+LKa-C=/Fk,2-KC`PMDJFR5=T5$'2"2"C2X/KB"2"CyN1-0+9/>C_5C("@9$'3"NE &IC+aB5-(HC%BAh'C%2:/K/>@9C@.!+.$'N0B"+C%2_/>C&I(5+(H$'N7A25$'&I-1,NÜ+.B"@øN0C%+M'@>(""C%+"CN1,@.M,C%B"@ø"Cå#N1-1XB"C<3*$'@92"!%CUÜÃ2 (HC%B5/+ a2X/."!/.-1+9C%@N1CB"@/.@>h,-1NG"C NbLdf %$'2+.B5-0h,2:/>C <$'[email protected]%#w5C%@.#w5CLB"2C2"+.C&3"N1C 5C ("@.$'([email protected]!=/>!+

P/>CNeXB"C N1C+("@9$'("@.-0!/.!%+áE,2"+

P"Cη(G)

cρ(G)

C/G⊕H

(HC%B5h,C%2:/e=/>@9C"!#%-0"!%C+"C("B"-1+áN1C+G(5@.$'("@9-1!=/>!%+åE2"+P"C

GC=/HUøCy(5@.$'3"N %&)C_+.B"-7h,2:/(*CB5/?/>@9C@.!%+9$'N1BC2/.C%&)("+(*$'N7a2"$'&)-b,NH(E,@D#C/./.C &)!/."$a"C,U

, 5 ?3

: HÂfZ¾°À5°ÁH Î%ÀEÎHfË pÁ

H øÌ .*),+ 68"B"2M'@.,(""CG = (V,E)

U 2 9')>45'N1C ("N0B"+ (HC/.-0/

r ∈ N/>CNåaB5C

V+9$'-0/y(E@9/>-7/>-0$'2"2E,35N1CC2

rC%25+.C%&3"N1C+-125"B"-1+.,2:/y"C%+M'@.,(""C+?+.,2"+

H-12""B5-0/

"$# ÌXZ¾%aÉkÈh;;()+(;/@CIH

KX1 Y?[:]\+^+D ?XD]Ik ∈ N

?XQcWBE/[L@CBD@? @EL]IDKH1 :?B YH/[a@CIC^H>D\+/B `5R13/>A@:?B?\+/KH[BED \m[/?OQB]>D 6 2 % 2 9>+ )>4 2 *+

H45 1 : 4;)0L]KH[BEDNL Y?[:]\+^+DNLFHD-B :?[YHD]KH[FHD a]B @BAQK+D

:?KA\mB K Lk*L]@B :

kR7D@?=\m[DNLNL]@/1 DNL]IyFH/1+13NDi:?b?DbaiBED-Y?[:]\+^+D?c

]>/1`L]I[:?I*@/1mc[Q,N0M'$'@9-0/>5&ICB5/>-0N1-1+9CNb&)!/."$a"C_"C)P EDJFR5%TVUtA$'-7/

G′ B52M,@>,("5C!/.-1XB"C/.!,UHÃ2 2"$,/.C_(E,@ Bq(G′)Nô[]C%2"+9C%&3"N0C_"C%+D+9$'B"+ö6YM'@.,(""C+?-02""B"-7/>+G"C

G′ Á!h,C%2:/>B"CN1N0C%&)C%2:/hA-0"C%+ Dmh,C%# ,B("N1B"+ q +.$'&)&IC=/>+UtA$'-7/q = |V (H)|

U[QN1M'$'@9-0/."&IC#,N0#%B"N0CI@9!%#%B5@.+.-7h'C&IC2X/%c(H$'B"@y#wE,XB"CI+.$'B5+6YM,@>,("5C`,("("Ü6@>-1+.+.,2:/E,2"+N1 5!%#%$,&I(H$'+.-7/>-0$'2`"BM'@>(""C,cENV[\C2"+.C&3"N0C_"C&B"N0/.-76gC%2"+9C%&3"N0C%+e+9B"-0h,2:/iS(G) =

⟨Bq(G[V1]), . . . ,B

q(G[Vr])⟩

: V1, . . . Vr

C+9/?B"2"Cy(E,@/>-7/>-1$,2`"C D .

øC%+#%N1,+.+9C%+?"CNb(E@9/>-7/>-0$'2V1, . . . , Vr

+9$'2:/?!h'C2:/>B"CN1N1C&IC2:/?ha-0"C%+UKB^O"2E,Nôc

VC+9/K(E,@9/.-0/.-1$'2"2",3"N1C_C2

r+.$'B5+6YC2"+.C&3"N0C%+-02""B"-0+>,2:/B"2ÆM'@.,(""C +>,2"+

H-02""B"-7/+.-C=/L+9C%B"N0C%&)C%2:/L+.--1NC A-1+9/.C«B52½&B5N0/>-Å6YC2"+.C&3"N1C

〈B1, . . . , Br〉"CS(G)

/>CN_aB5C«(H$'B"@ />$,B5/i ∈ 1, . . . r

cBi

2[]C%+/(",+ NV[\C2"+.C&3"N0C)#$'&)(*$'+9! B"2"-1XB"C&IC2:/ 5BkM'@.,(""C4ha-1"CcøC/y(H$'B"@_/>$'BA/H ′ ∈ Bi

cunlab(H ′)

2[]C%+/K(",+-1+.$,&I$'@9(""C_oHUÃ2(HC%BA/[email protected]& ,@9XB"C%@DXB[]-1N_ B"22"$'&3"@.CKO"2"-"C_M'@.,(""C+D!/.-1XB"C/.!%+D[\,B("N0B"+

q+9$'&)&IC=/>+%U

tA$'-7/l(q, k)

N1C`2"$,&3"@.C 5C M'@>(""C%+k6g!/.-1XB"C/.!%+4[\,B("N0B"+

q+.$,&I&)C/.+%U<WYN ,B("N1B5+

2l(q,k)C%25+.C%&3"N1C+(H$'+9+.-135N1C%+ø(H$'B"@Bq(G′)

cC=/"$'2"#(r+1)2l(q,k) &B"N7/>-Å6YC%25+.C%&3"N1C+º(H$'+.+9-13"N0C%+ f(H$'B"@º#wE,XB"CC%25+.C%&3"N1CcENb&B"N7/>-0("N1-0#%-0/.!C%+9/?#$'&I([email protected]+9C_C%2:/>@9C

0C/r U+,25+/.$'B"@DN0C%+#,+

r ≤ |V (G)|UEÃ_2 h,$'-0/

df,#-1N1C&IC2:/DXB"Cl(q, k)

C%+/3H$'@92"!(E,@kq × 2

q×(q−1)2UøC%+KC2"+.C&3"N0C%+

S(*CB5h'C2:/_/[email protected]#%,N1#B"N1!+@9!%#B"@.+9-0h'C&IC2:/U$'B"@N1C+$'(H!%@./>-0$'2"+

·cηC=/ρcH$'24-1&)&I!"-bÜ/>C%&)C%2:/§

S(·i) =〈∅, ·i, ∅, . . . , ∅〉

Á$∅@9C%("@9!%+9C%2:/>CN0C_M'@>(""C!/.-1XB"C/.!yha-0"C UEå2C *C/-1N2[]C=A-0+9/>CXB[]B"2"Cy(E,@/>-0/.-1$'2 5C%++.$'&)&)C/>+D5BLM'@.,(""C

·iU

§S(ρi→j(G)) =

⟨ρi→j(H

′) : H ′ ∈ Bi

⟩

i∈1,...,r

:⟨Bi

⟩i∈1,...,r

∈ S(G)

U§S(ηi,j(G)) =

⟨ηi,j(H

′) : H ′ ∈ Bi

⟩

i∈1,...,r

:⟨Bi

⟩i∈1,...,r

∈ S(G)

U$'B5@áN0Ce"[email protected]"-0C%@å#,+c'!/>,2:/á"$'2"2"!

S(G1)C/S(G2)

$'2 "$'-7/#,N0#%B"N0C%@S(G1⊕G2)

U $'B5/.C%+åN1C+(E,@/>-7/>-1$,2"+P = V1, . . . Vr

"CV (G1⊕G2)

("@.$mha-1C2"2"C2X/[]B"2"Cå(E@9/>-7/>-0$'2V ∩V (G1) : V ∈ P1"C

V (G1)C=/?[\B52"C (",@9/.-0/>-0$'2

V ∩ V (G2) : V ∈ P2"C

V (G2)UWYN(*CB5/f mh,$'-1@B"2½2"$'&[email protected] C=A(H$'2"C%2:/.-1C%N"C df$'2"+"C«#$'&3"-02"C%@ B52"C«(",@9/.-0/>-0$'2

P1

"CV (G1)

C/P2

"CV (G2)

cZ&I,-1+-1Nø25$'B"+?+.B /5C #$'2"2E 0/>@9CyN0C%+?C%2"+9C%&35N1C%+Bq(Gi[V

′])cHh'C#

i ∈1, 2

c5(H$'B"@DN0C%+D"-+º[email protected]/.C%+#%N1,+.+9C%+V ′ "C+?(",@9/.-0/>-0$'2"+ Pi

(*$'B5@D(*$'BAh'$'-0@e#,N0#%B"N0C%@eN0C%+DC2"+.C&3"N1C+Bq(G[V ′])

(H$'B"@DN0C%+#%N1,+.+9C%+V ′ "CN1 (",@9/.-0/>-0$'2`"C V (G)

U

MSOL

2LCºC=/cE$'2h'$'-7/Ddf,#%-0N1C&IC2X/eaB5CBq(G1 ⊕G2) = H1 ⊕H2 : H1 ∈ B

q(G1), H2 ∈ Bq(G2)

C/|V (H1)|+ |V (H2)| ≤ q.

tA$'-7/B1C/B2"CBAIC%2"+9C%&35N1C%+G"C?M'@>,(5"C%+G!/>-0XB"C/.!%+º[Q,BI("N1B5+

q+.$,&I&)C/.+%UaÃ2 "!=OE2"-7/(E@

B1 ⊕ B2NV[\C2"+.C&3"N0C

H1 ⊕ H2 : H1 ∈ B1, H2 ∈ B2C/|V (H1)| + |V (H2)| ≤ q

UÃ2u`5$'2"#Bq(G1 ⊕G2) = Bq(G1)⊕ Bq(G2)

US(G1 ⊕ G2)

(*CB5/=/>@9C`#,N0#%B"N0! "CINb^&I,2"- %@9CI+9B"-0h,2:/>C $'2k#%$'&)&IC2"#%C h'C%# NV[\C2"+.C&3"N0CD = 〈〉 × S(G1)× S(G2)

cZ("B"-0+$,2^!/.C%2"D

o Nô[]C%2"+9C%&35N1Cy"Cy/.$'B"+?N1C+?/[email protected](5N1C/.+aB5-(HC%BAh'C%2://[email protected]$'35/.C%2XB"+Kº[\B"2L/>@.-0("N1C=/(M,M1,M2) ∈ D

C%2^+.B"("([email protected]&I,2:/?B"2^C2"+.C&3"N0CB1

5CM1

C/B"2C%25+.C%&3"N1CB2

"CM2

cÜC/<C%2@.°89$'B5/>,2:/B1⊕B2

oMU'Ã2h,$'-0/XB"CeXBE,25

D2"C(H$'B"@9@>K(5N1B"+/[email protected]!/.C%2"5Bc'$'2 ,B5@>

S(G1⊕G2) = M : (M, 〈〉, 〈〉) ∈ DU l$'&I&)C

McM1

C/M2

,("(E,@/>-0C%2"2"C2:/o4B"2B"2"-0h,C%@9+[\,B("N0B"+

(r + 1)2l(q,k) !%N0!%&)C%2:/>+cDI,B("N0B"+

(r + 1)3×2l(q,k) !%N0!%&)C%2:/>+U2"C

k6YC=A("@9C%+9+.-1$,2

O(n)$'(*!@>/.-1$'25+

⊕c#wE,XB"CÄ$'(*!@>/.-1$'2 ("@.C2""+B"2 />C%&)("+

O((n +

1)3×2l(q,k))c!=/w,2:/"$,2"2"!uXB"CuNô[]$'(H!%@./>-0$'2

⊕+9B"@ N1C+`C%25+.C%&3"N1C+ +.CÆdf,-7/`C%2 /.C%&)("+ #%$'25+9/w2X/%cXBE,2"

qC/k+9$'2:/KO5A!%+Uº[]$'(H!%@>Ü/>-1$,2L(H$'B"@

·+.Cydf,-0/C2^/>C&I("+?#%$,2"+9/>,2:/C/#%CN1N0C%+(*$,B"@

ηC=/ρ+.C_dÁ$'2:/?C%2/>C&I(5+

O((n+ 1)2l(q,k))U+BO"2E,Nôc5N1C/>C&I(5+D"CNô[\,N1M'$,@.-0/."&)CC+9/?(*$,N<a2"$'&)-b,NVU

Ã2h,$'-0/aB5CáN0Cü("@9$'3"NE &IC<"C%&I,2"E2X/5C(E,@9/.-0/.-1$'2"25C%@ºNV[\C2"+.C&3"N0Cü"C%++9$'&I&)C/.+C%2V1, . . . , Vr/>CNaB5C^#wE,XB"C

Vi

-125"B"-0/ B52 M'@.,(""C+>,25+H1, . . .Hk

-12"5B"-0/)C%+/ !M:,N0C%&)C%2:/)(*$'N7a2"$'&)-b,NVcN1("@9C%B5h,C2[]C%+9/eaBº[\B"25CKN1!M %@9CK&)$A5-0OE#%/>-0$'24"CKN1y([email protected]%BAh'C("@9!%#!%"C2X/.C,UlC=/./.C#%N1,+.+9CK"C?("@.$,3"NE &IC+C%+/_C=5,#=/>C&IC2X/N1I#Nb,+9+.C"C+_("@.$,3"NE &IC+/>CN1+aBº[\-0N<C=A-0+9/>CB"2"CdÁ$'@9&B"N0Cϕ+.,2"+XBE,2:/.-0OE#%/>CB"@C/?"Ch,@.-1,3"N0C%+DN1-03"@.C+

x1, . . . xl

cHC=/?$LNV[\$,2L"C& 2""C"Cy(E,@9/.-0/.-1$'2"25C%@VC%2

V1, . . . , Vr

/.C%NXB"C(H$'B"@e/>$,B5/i ∈ 1, . . . , r

$'2,-0/〈G, Vi〉 |= ∀x1, . . . , xl ∈ Vi, ϕ(x1, . . . , xl).

Ã2Ih,C%@[email protected]"CNV[\$'2)(HC%B5/!/>C2""@9C#%C@.!%+9B"N0/>/G(*$'B5@G2[]-1&)(H$'@9/.C?XB"CN1N0C?dÁ$'@9&B"N0C?C2 N0$'M'-1XB"C?&)$'2EÜ6"-0aB5Cy5BL+9C%#%$,2"L$,@."@9C,Ukj7d 't Gfk_z

MSOL

lC%@/w,-02"C%+"!=OE2"-0/.-1$'25++.$'2:/"$'252"!%C+dÁ$,@.&)C%N1N0C%&)C%2:/C2y,2"2"C 5Cå"Cå#C#wE,("-7/>@.CU N1B"+ø"C5!/w-1N1++.B5@?N1/>"!$'@.-0C"C%+&)$A! N1C+eOE2"-1+D(HC%BAh'C%2:/K/>@9C_/>@.$,B5h'!E,25+ PåF [ÜTVU ,+1!14B $'$ 6 2 68D +M6"( $^22)¿*Í'À" pÂÁÀEf¾°È C%+9/GB"2)C%2"+9C%&35N1COE2"-E"CÎÉuHÂfÈaÎËÈ ¾°ÈaÂfÀ5mfHÎ ; UHlEaB5CK+@a&3H$'N0CD"[email protected]/.-1$'2 (*$'+9+7 "C B52"CÀ"¾ÜÁmÌZc"B52LC2X/.-1C@D2E/>[email protected]%Nø25$'22aB5NôUZ25Cy¾°ÈaÂÁÀ5°Á* [\,@.-7/>!

k ∈ N∗ +9B"@B"2C%25+.C%&3"N1CA

V,(5(*CN1!%C_!M:,N0C%&)C%2:/¾°ÈAÂÁÀ5°ÁHk>ÀEÁ¾°È +9B"@

A=cEC+9/?B"2+.$'B5+6YC2"+.C&3"N0Cy5C

AkU

Ð ! Ý ò°ÙVÒôÖ9æZßmäãÜÕ°Ó\ÙVÓQâ=ÕÜæEÝ=òmÙVâÒVÓëæVÖÕ%ÙZßmÖ9æEõâ=ÕÜæfÙôÝ=ÕÙôÖ9æ5ÖöÙHß°Ö9æ"÷ÅâÕÜõYÙôÓ\âÕÜæEß°Ý Õæ"×\Ö.æ"ó=âmõÝ=Þ°ò°×ëÝ ÓQÒVÖ.æÚ>ØDÝ Óëæ"Ö×Q×\Ö.æ"ûÖ9ò°ó=Ö9Õ%Ù ÙôÒVÖüÒôÖû°Òôä9æVÖÕ%Ùôä9æºûÜÝ Òß°Ö9æÒôÖ×ëÝwÙVÓQâ=Õæð

, 5 ?3

tA$'-7/τN1CKh,$A#%,3"B"N1,-1@9Ch'C##%$'&)&IC+@a&3H$'N1C+D"C%+D@9C%N1/>-0$'2"+

R1, . . . , Rk

c"[\,@.-7/>!%+a1, . . . , [email protected]+.(HC%#=/>-0h,C%&)C%2:/U2"C4Î=m¾pÍ,p¾°È

A"BÆh'$a#3"B"[email protected]

τf,("(HC%N1!y!%M',N1C&IC2X/KB"2"C

τ>Îm¾pÍ,p¾°È C%+/ü#$'2"+9/.-0/.B"!%Cá[]B"2C%25+.C%&3"N1C

Ac°N1CáËHÉkÀEfÈZc°C/[]B"2"C@9C%N1/>-0$'2

ai

6ö,[email protected]

i

+9B"@A(*$,B"@#wE,XB"C

i ∈ 1, . . . , kUEN0N1C+9C%@> 2"$,/.!%C

〈A,RA1 , . . . , R

Ak 〉U

BE,2"«-1N2[ «,B"@>(E,+y[\,&3"-0M'BV/>! +.B"@yNb +/>@.B5#/>[email protected]$'2«!%#@.-1@.(",@9dÁ$'-0+%cø(E@ 3"B"+y"C2"$,/>/>-0$'2c

RoÆN1Æ("Nb#%C`"C

RA U ,@4NbL+.B"-7/>C,c/>$'B5/.C%+ N1C++9/[email protected]"#=/>B"@9C%+ +.C@.$'2:/)+.B5("(*$,+.!%C+)O"2"-1C+f#[\C+9/6oÜ6g"-1@9C_"C"$'&I,-12"COE2"- U

KC%Ba4(H$'+.+9-13"-0N1-0/.!%+(*CB5h'C2:/å/>@9CeC%2:ha-1+.,M'!C%+(H$'B"@<@9C%("@9!%+9C%2:/>C@áB524M'@>,(5"C,U?,2"+üN1("@9C%&)-E @.C,cN1C+e!%N0!%&)C%2:/>+D+9$'2:/N1C+e+.$'&)&)C/>+"B M'@>(""C,c5C/D$'2 5-1+.(H$'+9C_[\B52"[email protected]/.-1$'2`35-12E,[email protected]?XB"-C%+/Dha@>,-0C+.-?N1C+I5C%BA +.$'&)&)C/>+)+.$,2X/`,Ü8.,#C%2:/>+U 2"+ N1Ä+9C%#%$,2""C,cGN1C+ !%N0!%&)C%2:/>+)+.$,2X/ N1C+ +.$,&I&)C/.+IC=/N1C+y,@.=/>C%+y"BÄM'@>(""C,UÃ2Ä"$'-0/5$'2"#)°89$'B5/.C%@yB"2Ä("@.!"-1#%/y(*$'B5@y+.h'$'-0@y+9-B"2k!N1!&IC2X/yC+9/ B"2+.$,&I&)C/%c<$'BB"2"C,@9/.C,UáÃ2Ïu/.$'BÜ89$'B5@.+4B"2"C@9C%NbÜ/>-1$,2 3"-12",-1@9C,c<&I,-0+#%C=/./.CdÁ$,-1+#-DNbÆ@.CNb/.-1$'2[]-12"#-1"C2"#%CC2X/[email protected]++.$'&)&IC=/>+DC=/N0C%+,@9/.C%+%UÃ2u"!%+9-1M'2"C (E,@τ1N1Cyh'$a#,35B"Nb,[email protected](*$,+.+.!E,2:/B"2Æ+9C%B"N<+@a&3H$'N1C "[email protected]/.-1$'2

Adj[\,@.-7/>!

2UÃ2 "!+.-1M,2"C`(E,@

τ2N0C)h,$A#%,3"B"N1,-1@9CI(H$'+9+.!%",2:/ B"2 +@a&3H$'N0C "C`@9C%N1/>-0$'2

Adj2[\,@.-7/>!

2C=/4B"2+@a&3H$'N0C_"[email protected]%N1/>-0$'2

PV

[\,@.-7/>!1U

tA$'-7/G = (V,E)

B"2ÄM'@.,(""CUÃ_2Ä2"$,/.C%@.τ1(G)

Nbτ16g+9/>@9B"#/.B"@.C

〈V,Adj〉cø$

Adj"!+.-0M'2"CNô[\,Ü8.#%C%25#%CC%2:/>@9C^N1C+I+9$'&)&IC=/>+%cåC=/

τ2(G)Nbτ26g+9/>@9B"#/.B"@.C

〈V ∪ E, PV ,Adj2〉$e

PV

C+9/ B"2("@9!%"-0#/ B"2E,[email protected] ha@>,-á+.-åNô[]!%N0!%&)C%2:/ C%+9/E,2"+VcC/

Adj2C%+9/B"2"[email protected]%N1/>-0$'2Ä3"[email protected]@>-1C)+.-áB"2!%N0!%&)C%2:/C%+9/B"2+9$'&)&IC=/cmNô[\,B5/[email protected]åC%+/<B52"Ce,@9/.CGC=/<N0CG+9$'&I&)C/C+9/-12"#-1"C2X/ oNô[\,@.=/>C,UÃ225$,/>CáXB"CNb+9/>@9B"#/.B"@.C

τ2(G)@9C%+/>@.C-12:/>C"C

PV

C%+9/ü-1+.$,&I$'@9(""Cyoτ1(G)

c,$I(G)

C%+/åN1CM,@>,("5CD[\-02"#%-0"C%25#%C"CGU

[\,B5/>@9C%+D$'3X89C/>+D&I/."!%&I/>-0XB"C%+G(*CB5h'C2X/=/>@[email protected]%(5@.!%+9C%2:/>!+(E,@e"C+D+9/[email protected]"#=/>B"@9C%+%c5#%$'&)&ICN1C+M'@.,(""C+4$'@.-0C%2:/>!+ fNbÆ@9C%N1/>-0$'2k2[]C%+/)(E,+4+@a&)!/[email protected]"C C/ N0C%+4[a(*[email protected]'@.,(""C+ Á(E,@4N0C%B"@M,@>,("5C[]-12"#-1"C2"#%C =U #, 4$ . ,9 #&/4 $ & 2 $+-,/9#& ,%&/%$2uN1$'M'-0XB"C 5Bu("@.C&I-0C%@?$'@."@9C,c*N1C%+XBE,2:/>-7OE#/.C%B"@9+(H$'@9/.C%2:/_+.B5@N1C+K!%N0!%&)C%2:/>+K"BÆ"$'&I,-02"C,U

2N1$'M'-0XB"C5B+9C%#$'2"$'@[email protected],cA$'2 (HC%B5/DXBE2X/.-0OEC@D!%M:N1C%&)C%2:/+.B"@DN0C%+@9C%N1/>-0$'2"+%UAN1$'M,-1XB"C?&I$62E,5-1XB"C"Bu+9C%#$'2"Ä$'@."@9C MSOL

(H$'B"@$>/1m:MF?@ aiLDbaN/1mF /[FHD][BE/7Y?@ a C+9/B"[email protected]%+/>@9-1#/.-1$'2L"CNb4N0$'M'-1XB"CK"B^+9C%#%$,2"^$'@9"@.CcE(*C@.&)C/9/w,2:/?"CXBE,2:/>-7OEC%@?B"2"-0aB5C%&)C%2:/?+.B"@[email protected]%N1/>-0$'2"+DB"2",-1@9C%+%U,25+Nb)+.B"-7/>C,cH+>2"+-125"-1#%/>-0$'2"+?+.B"("(5N1!%&)C%2:/>,-1@9C%+%cHN1C+dÁ$'@9&B"N0C%+?+.C%@9$'2:/+.B"(5(*$'+9!%C+_/>@9C&)$'2EÜ6"-0aB5C%+"B+.C%#$'2"L$'@."@9C,U2"CGdÁ$'@.&B5N1CáC%+9/üÀ5mHÉ ;pÈ_+9-ACN1N0CGC+9/"CNb?dÁ$,@.&)C

x1 = x2

$'BR(x1, . . . , xa)

c$x1, . . . , xa+.$,2X/"C+øh,@.-b3"N1C+ø"B("@.C&I-0C%@ø$'@9"@.CcC/

RC+9/B"2+@a&3H$'N0C"Cå@.CNb/.-1$'2_[\,@9-0/>!

a"Bh'$a#,3"B5Nb,[email protected],c$'BB"2"Ch,@.-1,3"N0CK"B+.C#%$'2"L$'@[email protected],U2"C ¿"À5¾ÁÀ"ÂfÈ ÂVÁ¾°ÈL[\B52"C)dÁ$,@.&B"N0C4C%+/B"25C)h,@9-b,3"N0C4XB"-å2[\C+9/ (E,+yE,2"+yNb (H$'@9/.!%C []B"2XBE,2:/>-7OE#Ü/>C%B5@%UZ25CdÁ$,@.&B"N0C_C%+/KÍ:ÂÁHÎ%È +9-øC%N0N1C2[\),B"#%B52"C_h,@.-b3"N1CN0-13"@9C,U2"CÀEÎÎzHÀA°fHu(H$'B"@B52 C%25+.C%&3"N1C"Ch,@9-b,3"N0C%+

XC/B"2"C+9/>@9B"#/.B"@.C

AC%+/eB"2"C?dÁ$'2"#=/>-1$,2XB"- o #wE,XB"C4h,@.-1,3"N0CN1-03"@.C"B ("@.C&I-0C%@$'@9"@.C "C

X,+.+9$A#-1C4B"2 !N1!%&)C%2:/"B "$,& ,-02"C"C

AcC/io`#w",XB"Ch,@.-1,3"N1C N1-03"@.C "Bu"C%Ba5- %&)C4$'@[email protected]"C

X,+9+.$a#%-1CB"2 +.$'B"+ö6YC2"+.C&3"N1C45Bu"$'&I,-12"C

MSOL

"CAUZ25C

τ6Y+/>@.B5#/>[email protected]

Ah'C# B"25C,+9+.-0M'2E/.-1$'2

z"C%+h,@.-1,3"N0C%+

X1, . . . , Xl

C+9/K,+.+9-1&)-1Nb3"N1C-oB"2"C(τ ∪ X1, . . .Xl)

6Y+/>@9B"#/.B"@.CUtA$'-7/

ϕB"2"CDdÁ$'@.&B"N1CU,?2"C

τ6g+9/>@9B"#/.B"@.C

Ah'C%#B"25C+.+.-0M'2E/.-1$'2"C%+h,@.-1,3"N0C%+üN0-13"@9C%+ü"C

ϕC+9/B"2ÄÉkøË%aÂfÈ"C

ϕ+9-ϕC+9/+>Ü/>-1+df,-7/>C",2"+_NbI+9/>@9B"#/.B"@.C

Ah'C#4Nô[\,+.+9-1M'2"/>-0$'2

z"C+h,@.-b3"N1C+

fN14"!OE25-0/>-0$'2 dÁ$'@.&)C%N0N1CC+9/"$'2"25!%CyC%2Æ2"2"C=AC =U*Ã2^N1C_25$,/>C@>〈A, z〉 |= ϕ

cE$'BA |= ϕ

+.-ϕC%+9/#%N0$'+.CU`Ç¾°gHËÈHp¾ÆËÈ ;pÀ"*° Í'À5mfHýº[\B"25CdÁ$'@.&B5N1C ϕ cº2"$,/.!%C qr(ϕ)

cC%+/"!O"2"-1C@9!%#B"@.+9-76h'C&IC2:/UtA-ϕ2[\`(E,+"C4XBE2X/.-0OE#%/>CB"@%cº,N1$,@.+

qr(ϕ) = 0U tA-

ϕ = ¬ψcøN1$'@9+

qr(ϕ) = qr(ψ)U

tA-ϕ = ψ1 ∧ ψ2

cZ$'Bϕ = ψ1 ∨ ψ2

cº,N0$'@.+qr(ϕ) = max(qr(ψ1), qr(ψ2))

U_å2AOE2Æ+.-ϕ = ∀Xψ

$'Bϕ = ∃Xψ

Á$XC%+9/B"2"C_h,@9-b,3"N0C"B("@.C&I-0C%@e$,BL"B+9C%#%$,2"^$'@9"@.C =c

qr(ϕ) = 1 + qr(ψ)U

tA$'-7/τB"2 h,$A#%,3"B"N1,-1@9C,UÜÃ22"$/>Cá(E,@

MSOLq(τ,X,Y)NV[\C2"+.C&3"N1Cá"C+dÁ$'@9&B"N1C+&I$'2","-1XB"C+"B +9C%#$'2" $'@9"@.CL"B h'$a#3"B"[email protected]

τcá"CL("@.$,dÁ$,2""C%B5@ "CÆXBE2X/.-0OE#%/>-0$'2 B ("N0B"+

qcGmh,C%#Æ5C%+h,@.-1,3"N0C%+"B ("@9C%&)-1C@_$'@."@9C4E,2"+

XC=/N1C+h,@.-b3"N1C+"BÄ+.C%#$'2"Ä$'@[email protected],25+

YUÃ2 2"$,/>C4(",@

MSOLq(τ)NV[\C2"+.C&3"N0C"C%+ødÁ$'@9&B"N1C+º&)$'2E,5-1XB"C%+ø5By+9C%#$'2"y$'@."@9C#%N0$'+.C+ø"B_h'$a#,3"B5Nb,[email protected] Á#,[]C%+/96

oÜ6g"-1@9CMSOLq(τ, ∅, ∅)

=U # ÌXZ¾ÜfÈ"C〈A, z〉

cH2"$,/.!Th(〈A, z〉)

cºC+9/NV[\C2"+.C&3"N1C"C%+dÁ$'@9&B"N0C%+K5$'2:/〈A, z〉

C%+9/B52&)$A! N1Cc5#,[]C%+/96oÜ6Y"[email protected](〈A, z〉) = ϕ : ϕ ∈ MSOL(τ)

C/〈A, z〉 |= ϕ.

KC&)%&)CTh(A) = ϕ : ϕ ∈ MSOL(τ)

C=/A |= ϕ.Ã225$,/>C@>

Thq(〈A, z〉)ca$

k ∈ Nc5Nô[]C%25+.C%&3"N1C5C%+edÁ$'@9&B"N1C+e"C_("@9$,dÁ$'2"5C%B"@e5C_aB",2:/>-0O"#Ü6/>-0$'2Ä,BÄ("N1B"+

q"$,2X/

〈A, z〉C%+9/ B"2Ä&I$a! N1C,cº#,[]C%+9/6oÜ6g"-1@9C

Th(〈A, z〉) ∩MSOLq(τ)UWKC)&)%&)C

Thq(A) = Th(A) ∩MSOLq(τ)U

lC%@/w,-02"+,N0M'$'@9-0/>5&IC+& 2"-1("B5N1C%2:/y"C++.$,B"+6gC%2"+9C%&3"N0C%+5CMSOLq(τ)

U $'B"@ #%C%N15cø-1Ndf,Ba6"@.,-0/?XB"Cy#%C/C%25+.C%&3"N1Cy+.$'-7/OE2"-ôcZ#C XB"-2[]C%+/(E,+?N1C#%,[email protected]#/.C%&)C%2:/cZ&I,-1+DXB"-Nô[]C%+/&)$a"B"N1$Nô[]!%XB"-7h,N1C2"#%C_"C+DdÁ$'@.&B5N1C%+UJ^,N0"C%B"@9C%B"+9C%&)C%2:/cNV[\!aB5-0h,N1C2"#%C^C%2:/>@9C«N0C%+`dÁ$'@9&B"N1C+ f&)%&)CÆ"B("@9C%&)-1C%@ $'@."@9C C+9/-12A6"!#%-1",3"N1CU <,@)#%$,2X/[email protected]$'2ý(*CB5/)/>@.$,B5h'C@IB"25CL@9C%NbÜ/>-1$,2Ïº[\!aB5-0h,N1C2"#%C f#[\C+9/6oÜ6g"-1@9C@9!EC A-0h'Cc+@a&)!/[email protected]"C C/?/>@.,2"+.-7/>-7h'C cZ2"$,/>!C≈cHC%2:/>@9CN1C+?dÁ$,@.&B"N0C%+/>CN1N1CXB"C +.-

ϕ ≈ ψcH,N1$'@9+

ϕC%+/!aB5-76h,N1C2:/>Cgo

ψcøC/ (*$,B"@B"2

q ∈ NOA5!c-1NÄ,-7/B"2k2"$,&3"@.C4OE25-á5C)dÁ$'@9&B"N0C%+2"$'2Ä!%XB"-7h,N1C%2:/.C%+(E,@

≈c"5C_("@.$,dÁ$,2""C%B5@D"C_XBE,2:/>-7OE#/.-1$'2B ("N1B"+

qU"4"!=OE2"-0/.-1$'2 "C_#%C=/./>C@9C%NbÜ/>-1$,2c5-12"+9-ºXB"CNb3H$'@.2"C+.B"(H!%@9-1CB"@.C"B`2"$'&3"@.C"CK#Nb,+9+.C%+G[]!%XB"-0h,N0C%2"#CK(H$'B"@GN1C%+ådÁ$'@9&B"N1C+G5CK("@9$,dÁ$'2""CB"@á"CXBE,2:/>-7OE#Ü/>-1$,2LB(5N1B"+

q+9$'2:/5$'2"2"!C%+?C%2L,2"2"C 5CU

,2"+ Nb«+9B"-0/.C,cá$'2 &I,2"-0("B"N0C%@> N1C+IdÁ$'@9&B"N0C%+ &I$a"B"N0$≈cGC/`5$'2"#^$,2 +.B"(5(*$'+9C%@.kXB"CÆN1C+C%25+.C%&3"N1C+

MSOLq(τ)+9$'2:/OE2"-1+c"(H$'B"@

qC/τO5A!+%U

,0#"4<*g6*g2MSOL

$'B5@?+9-1&)("N1-7OEC%@N1C%+D25$,/w/.-1$'25+%c"+9-XC+9/?B"2"[email protected]/.-1$'2 B"2E,[email protected],c5$'2L!#%@9-1@>

x ∈ X(H$'B"@

X(x)UÃ2L!#%@9-1@>

∀x ∈ X,ϕ(H$'B"@

∀x, x ∈ X ⇒ ϕC/∃x ∈ X,ϕ

(H$'B"@∃x, x ∈ X ∧ ϕ

U

, 5 ?3

Ì ÁmfH ;;(;(*)+(25C("@9$'("@.-0!/.!G+9B"@<N0C%+ τ 6Y+/>@.B5#/>[email protected]%+ P C%+/üB"2"CÇ¾°ZÇ¾ÜÁÌ:mÌ MSOL+%[]-1NXC=A-1+/>CB"2"C4dÁ$'@9&B"N0C&)$'2E,"-0XB"C"B +.C#%$'25«$'@9"@.C4+9B"@N1Ch,$A#%,3"B"N1,-1@9C

ϕ/>CN1N0CXB"C

PC+9/_ha@>,-0C4+.B"@_N1+9/[email protected]"#=/>B"@9C

A+.-øC=/?+.C%B5N1C%&)C%2:/?+.-

A |= ϕÃ2L(HC%B5/,(5("N1-0aB5C%@eN1 5!OE2"-7/>-0$'2 +.B"@N0C%++9/[email protected]"#=/>B"@9C%+τ1C/τ2"C+M,@>,("5C%+%UHÃ2$'35/.-1C%2:/

Ì ÁmfH ; (;(;(_2"CL("@.$,("@.-0!/>!+.B5@ N1C+IM,@>,("5C%+ C+9/ MSOL1

+%[]-1NC=A-1+/>CLB"2"CLdÁ$,@.&B"N0CC%2N1$,M'-1XB"C_&)$'2E,5-1XB"C"B+.C%#$'2"Æ$,@."@9Cϕ+.B"@?N1CN1,2"M:,M,C

τ1/>C%N0N1C_XB[]B"2^M,@>,("5C,-7/Nb4("@9$'("@.-0!/.!+.-øC=/?+.C%B5N1C%&)C%2:/?+.-

τ1(G) |= ϕU2"Cy("@9$'("@.-0!/.!y+9B"@?N1C%+M'@.,(""C+C+9/

MSOL2

+[\-0NøC=A-1+/>CyB"2"CdÁ$'@.&B5N1C_&)$'2E,"-0XB"C_"B^+9C%#$'2"$'@9"@.Cϕ+.B"@N0C_Nb,2"M',M'C

τ2/.C%N1N0CXB[]B"2^M'@.,(""C-0/DN1 (5@.$'("@9-1!=/>!_+.-øC=/?+.C%B5N1C%&)C%2:/?+.-

τ2(G) |= ϕU

<,@C 5C&I(5N1C,cEN1("@.$,("@.-0!/>!GC%+9/?#$'2"2"C 5C C%+/

MSOL1

cEh,C%#ϕ , ∀X, (∀x ∈ X, ∀y,Adj(x, y)⇒ X(y))⇒ ∀x, x ∈ X.

4("@.$'([email protected]!=/>!G(HC%BA/=/>@9Cy#$'N1$,@.-1!mh,C%#

3#%$'B"N0C%B"@9+ Nô[]C%+9/?!M:,N1C&IC2:/_

ϕ , ∃X∃Y ∃Z, Stable(X) ∧ Stable(Y ) ∧ Stable(Z) ∧ Partition(X, Y, Z)

h'C#Partition(X, Y, Z) ,∀x, (x ∈ X ∨ x ∈ Y ∨ x ∈ Z) ∧ ¬(x ∈ X ∧ x ∈ Y )∧

¬(x ∈ X ∧ x ∈ Z) ∧ ¬(x ∈ Y ∧ x ∈ Z)C/Stable(X) , ∀x ∈ X, ∀y ∈ X, x 6= y ⇒ ¬Adj(x, y).^("@.$,("@.-0!/>!G(H$'+.+ %"C B"2 #-1@9#%B"-7/ ,&I-0N0/.$'2"-1C2«C%+/

MSOL2

UÃ2 #w"C@.#w"CB"2 +9$'B"+ö6C%25+.C%&3"N1C[\,@.=/>C+X/>CNøXB"C

(V,X)+9$'-0/

26g@.!M'B"N1-0C%@eC=/#%$'252"C=AC,U

ϕ , ∃X ⊆ V,

[[∀x ∈ V,∃e, f ∈ X, (e 6= f ∧ Adj2(x, e) ∧ Adj2(x, f)∧

∀g ∈ X,Adj2(x, g)⇒ (g = e ∨ g = f))]∧

[∀V ′ ⊆ V,[∀u ∈ V ′, ∀v ∈ V, (∃e ∈ X, (Adj2(e, u) ∧ Adj2(e, v)))⇒ y ∈ V ′)

⇒ V ⊆ V ]]].

Ã2 h'$,-0/df,#%-0N1C%&)C%2:/XB[]B"2"CG("@.$'([email protected]!=/>!MSOL1

(HC%B5//[email protected]å/>@>"B"-0/.CGC2B"25CG("@9$'("@9-1!/.!MSOL2

UøC#$'2:/>@>[email protected]á2[]C%+/(E,+/>$'BÜ89$,B"@.+hA@.,-mN1("@9$'("@.-0!/.!G(H$'+.+ %"CB"24#%-0@.#%B5-0/ K,&)-1N7/>$'25-1C%2 2º[\C+9/(E,+

MSOL1

Pl$,B,,TVUlC%@/w,-02"+("@9$'3"NE &IC+; <6g#%$'&)("N1C=/>++.$'2:/"$'25#

MSOL1

$'BMSOL2

U ,@C=AC&I("N0C,c*N1C%+K("@.$63"N %&)C%+k 6 2 % 2 98+-)>4 2 C=/ @ + 9,)>4;)>4 2

k.!/"$#0%, 132 "$45>+ .)> f(H$'B"@

kOA5! +.$'2:/

MSOL1

UEøC%+?("@9$'3"NE &IC+ 68459&68:.4;) + "$45% ) 2 .45«C/ 2 :.AC98+ @/<. 68: #04 :.L+.$'2:/

MSOL2

UøC%+(5@.$'3"N %&)C%+

LinEMSOL+9$'2:/eB"2I!%XB"-0h,N0C%2:/G"C%+e(5@.$'3"N %&)C%+

MSOL(H$'B"@áN1C+G("@9$'3"N %&)C%+[]$'(5/.-1&)-1+>Ü/>-1$,2"+%U

MSOL

Ì ÁmfH ;;(E;( ;(KPulDJHG?R'RT2 ("@9$'3"NE &IC[]$'(5/>-0&I-0+>/.-1$'2)+.B5@eNb#Nb,+9+.C5C%+ τ 6g+9/[email protected]"#=/>B"@9C%+DC%+/B"2("@.$,3"NE &ICLinEMSOL

+[\-0N(*CB5/?/[email protected]=A("@9-1&)!_+.$'B"+?NbdÁ$,@.&)C_+.B"-7h,2X/.C9 /w,2:/"$'2"25!ÄB52"C

τ6Y+/>@9B"#/.B"@.C

AceC=/

mdÁ$'2"#=/>-0$'2"+ "C (H$'-1"+

f1, . . . , fm

"!=OE2"-1C++.B"@Ac/>@9$'B5h,C%@B"2L,+.+9-1M'2"C&IC2:/

z"C+?h,@9-b,3"N0C%+eN0-13"@9C%+e5C

φ/.C%NøXB"C

∑

1≤i≤i1≤j≤m

aij|z(Xi)|j = opt

∑

1≤i≤i1≤j≤m

aij|z′(Xi)|j : 〈A, z′〉 |= ϕ(X1, . . . , Xl)

$ϕC+9/B"25CKdÁ$'@.&B"N1C :,2:/#%$,&I&)Ch,@9-b,3"N0C%+á[]C%2"+9C%&3"N0C%+DN0-13"@9C%+

X1, . . .Xl

copt

C+9/+9$'-0/min$'B

maxcaij

C%+/B52LC2X/.-1C@?(H$'B"@e/.$'B5/1 ≤ i ≤ l

C=/1 ≤ j ≤ m

cEC/|z(Xi)|j =

∑a∈z(Xi)

fj(a)U

Ã2"-0@>DXB[\B52("@9$'3"NE &ICå+9B"@N0C%+M'@.,(""C+C+9/B"2("@9$'3"N %&)CLinEMSOL1

+[\-0N:+%[\,M'-0/[]B"2 ("@.$63"N %&)CLinEMSOL

+.B"@GNb +/>@9B"#/.B"@.Cτ1"B`M,@>,("5C,caC/eXB[]-1NH+%[\,M'-7/á[\B52 ("@.$'35NE %&)C

LinEMSOL1+%[]-1N?+%[\,M'-7/ º[\B"2 (5@.$'3"N %&)CLinEMSOL

+.B"@ Nb«+/>@9B"#/.B"@.Cτ2"B M'@.,(""CUøC%+ ("@.$'35NE %&)C%+

!"$#&%'*),+ #&%'c !"$#&%' 1 2 "$45>+3>)üc- 1 #/+86 9')> ! )+9$'2:/L5C%+("@.$,3"NE &IC+LinEMSOL1

U ,@#%$'2:/[email protected]_("@9$'3"N %&)C 2 "$#09> 68<.9 2 "J+ )>4 : 2"CNô[]C%+/?(E,+yP\,B 'SÜTVU"$# ÌXZ¾%aÉkÈ ;;(E;(y, 8W /K 76_;/@CI

k ∈ ND]I

ΠKH1\m[N/?OQB]>D

LinEMSOL2

?XQc;BE/[LΠ\+D]KHI ]I*[D$[NL/B K0D]10ID]>\+L B @C137:?@C[DiL]KH[BEDNL Y?[:]\+^+DNLFHDB :?[*YHD]KH[A:?[=O]/[DNLHaND]1+ID$O]/[a13ND\`:?[

k*L]@

B :GFHbaN/>\+/L]@CI@/1:?[=O]/[DNLHaND]1+ID DNL]I FH/1+13ND?c"$# ÌXZ¾%aÉkÈ ; ( ;( , . 2M26;/@CI

k ∈ ND]I

ΠKH1 \m[/?OQB]>D

LinEMSOL1

?XQcHBE/[LΠ\+D]KHI]I*[D

[NL/B KJD]1JID]>\+LB @C137:?@C[NDyL]KX[BEDNLY?[:]\+^+DNL-FHDB :?[YHD]KH[FHD/a]B @BAQK+D-O]/[a13ND\`:?[k*L]@;B :

kR7D@?=\m[DNLNL]@/1

F?KY?[7:]\+^+D$DNL]I FH/1+13ND?c

kj « xoÁxoVtuvMSOL

,2"+#%C=/./>CG+.C#/>-0$'2cm$'2 (E,@9N1C%@.?º[\B"25C2"$'BAh'C%N0N1Cå#%N1,+.+9CG5CG("@9$'3"N %&)C%+%U°WYNX+%[\,M'-7/"C%+("@.$,3"NE &IC+"C (E,@9/.-0/.-1$'2"C+K+.$,&I&)C/.+[]B"2ÆM,@>,("5C Á$'BÆ("N0B"+M,!%2"!@>,N0C%&)C%2:/KN0C 5$'& -12"C[]B"2"C +9/>@9B"#/.B"@.C C%2V1, . . . , Vr

/>CNøaB5Cy#w",XB"C #Nb,+9+.C"CNb4(E@9/>-7/>-0$'2 ,-7/DB"2"Cy("@.$,("@.-0!/>!MSOL

U Ì ÁmfH ;;( ()+(?2 ("@9$'3"N %&)C Π

C+9/)B"2ýÇ¾°Z<Â%aÉkÈ ËÈ ÇÀ"¾m°r°ÁHÈaÉkÈa*MSOL

+[\-0NC=A-0+9/>C4B52"CdÁ$'@.&B"N1Cτ&I$'2","-1XB"C"B +.C%#$'2" $'@9"@.C

ϕh,C%# B"2"Ch,@9-b,3"N0C[]C%25+.C%&3"N1C4N0-13"@9C

X/>CN1N0CXB"CΠ(HC%BA/=/>@9CyC A("@.-0&I!+.$,B"+NbdÁ$'@9&IC_+9B"-0h,2:/>C9

/w,2:/D"$,2"2"!B"25Cτ6g+9/>@9B"#/.B"@.C

AC=/DB"2 C2X/.-1C@

rcAN0CK"$,& ,-02"C?"C

A(HC%B5/6Y-1NH/[email protected](",@9/.-0/>-0$'2"2"!C%2

A1, . . . Ar"C_df$'2aB5C

∀i ∈ 1, 2, . . . r, 〈A, Ai〉 |= ϕ(Ai)

Ã2"-7/XB[]B"2 ("@.$'35NE %&)Cå"CáM'@>,(5"C%+C%+/üB"2y("@.$,3"NE &ICå"Cá(E,@/>-0/.-1$'252"C%&)C%2:/MSOL1

+[\-0N:+%[\,M'-0/[]B"2 ("@9$'3"N %&)C)5CI(E@9/>-7/>-0$'2"2"C&IC2X/MSOL

+.B"@yNb+9/[email protected]"#=/>B"@9Cτ1"BkM'@.,(""CcC/B52k("@9$'3"NE &IC"C (E,@/>-7/>-1$,2"2"C%&)C%2:/

MSOL2

+%[]-1N+[QM'-0/[]B"2½("@.$'35NE %&)CÆ"CÄ(E,@/>-7/>-1$,2"2"C%&)C%2:/MSOL

+.B5@N1+9/[email protected]"#=/>B"@9Cτ2"BLM,@>,("5C,U

, 5 ?3

<,@C 5C&I(5N1C,cDN0C%+ ("@9$'3"N %&)C%+ 6 2 % 2 98+-)>4 2 c 6 2 % 2 98+-)>4 2 *+3.H

45 1 : 4;)üc 2 "$#&9>132 "J+-)>4 :.øc8@ + 9,)>4;)>4 2 AC98+3@<. + 36 :.. @9 2 @/9>4 )

ΠÁ$

ΠC%+9/ B"2"C ("@9$'("@9-1!/.!

MSOLºC=/ @ + 9,)>4;)>4 2 AC9>+ @< @ +39 +34;)>ü+.$,2X/5C%+("@.$,3"NE &IC+"C(E,@/>-7/>-1$,2"2"C%&)C%2:/

MSOL1UøC%+e(5@.$'3"N %&)C%+ 6 2 % 2 98+-)>4 2 1 +39 )>KC/@ +39')>4;)>4 2 . "$.) AC98+3@<./ + "$45% ) 2.45/. +9$'2:/5C%+("@.$,3"NE &IC+D"C(E,@9/.-0/.-1$'2"25C%&)C%2:/

MSOL2UÃ2Æ(*CB5/@.C%&I,@9aB5C%@KXB"C +.-$'2LOA5C

rcZN0C ("@9$'3"N %&)CyC+9/KB"2Æ("@9$'3"NE &IC

MSOLcHh'C##%$'&)&)CdÁ$'@9&B"N1C

ψ = Partition(X1, . . . , Xr) ∧ ϕ(X1) ∧ . . . ∧ ϕ(Xr)U

$'B5@@.!+.$'B"[email protected]`C2k/>C&I("+ (*$,N<a2"$'&)-b,N<B"2 ("@.$,3"NE &IC)"C (",@9/.-0/>-0$'2"2"C&IC2:/cø-1Nå2"C +.B /4(E+[]B5/>-0N1-0+.C%@NV[Q,N0M'$'@9-0/>5&IC5C l$'B"@9#%C%N0N1Cc<JLK,$+ KX-DC=/ G$,/>-0#%+y(*$'B5@N1C+ (5@.$'3"N %&)C%+MSOL

(*$,B"@rC%2:/[email protected]

1C=/|V |c#%,@Nb#%$'2"+/w,2:/.C«"B/>C&I("+[]C=A!%#B5/>-0$'2 5CkNô[\,N0M'$'@.-7/>"&)C"!%(HC%2" "C«N1dÁ$'@9&B"N1CU <,@IC=AC%&)("N0C,c

k +868<.9 2 "J+ )>456 .: "$#09cá(H$'B"@

kO5A!,cGC%+/`B"2 (5@.$'3"N %&)C

MSOL1

c&I,-1+ + 68< 9 2 "J+-)>4 :. :."$#&9ÄC+9/ ; <6Y#$'&)("N1C=/D+.B"@N0C%+#%$'M,@>,("5C%+yP G$A8 ÜTVU 3$ (f&1$ + 4#$ &19$ 2 68D#+M6"1 $Nb,@9M'C%B"@G"C#N1-1XB"CK(*CB5/e/>$,B5/e2EÜ/>B"@9C%N1N0C%&)C%2:/e=/>@9C_!/>C2""B"C o/.$'B5/.C%+eN0C%+e+/>@.B5#/>[email protected]%+U $,B"@#%CNb^$,2 Æ3*C+.$'-02 []B"2"C`$,(*!@>/.-1$'2

ηR(*$'B5@#wE,XB"C + A&3*$'N0C "C @.C%N1/>-0$'2

RU<Ã2L/>$,BÜ89$'B"@9+B"2"C $,(*!@>/.-1$'2

ρ(H$'B"@N0C`@.C 6Y!=/>-1XB"C=/w,M,C,c<B52"C $'(H!%@>Ü/>-1$,2

⊕(H$'B"@NV[\B"25-1$'2cC/

·[email protected]("@.!+.C%2:/.CLN1+9/[email protected]"#=/>B"@9C º[\B"2!%N0!%&)C%2:/!/.-1XB"C=/>!

iUZ4"!O"2"-0/.-1$'2C%+/?"$'2"2"!C C2^2"2"C=ACUWYNZC%+9/Gdf,#%-0N1Coh'$'-0@áXB"CNbNb,@9M'C%B5@"CK#N1-1XB"C[]B"2`M,@>,("5C

GC+9/!%M:,N0Co Nb_N1,@.M'CB"@å"C?#%N1-0XB"C"C Nb

τ16Y+/>@9B"#/.B"@.C

τ1(G)UMå2ÄC *C/%cø#%$'&)&)C

Adj(v, v)C+9/df,BA (H$'B"@/.$'B5/

v"B "$'&I,-12"Ccº$'22"C4(HC%B5/(",+h'$'-0@Nô[]$'(H!%@>Ü/>-1$,2

ηAdj(i,i)

f(H$'B"@_B"2iE,25+

1, . . . , kE,25+yN1 #%$,2"+9/[email protected]"#=/>-1$,2Ä5C4Nb+9/[email protected]"#=/>B"@9C

τ1(G)U KC4("N1B5+K#%$,&I&)[email protected]/.-1$'2

AdjC+9/_+ a&I!=/>@.-0XB"C,c*$'2u(*CB5//.$'BÜ89$'B"@9+/.@.$'B5h,C%@B"2"CIC=A("@9C%+.+9-1$'2k(*$'B5@

τ1(G)/>C%N0N1C XB"C NV[\$,(*!@>/.-1$'2

ηAdj(i,j)

+9$'-0/y+9B"-0ha-1C)"CINô[]$'(*!@>/.-1$'2ηAdj

(j,i)

U?2"Ck6gC=A("@9C%+.+9-1$'2(H$'B"@

G("@9$°ha-0C%2""@.,-0/[email protected]#/.C%&)C%2:/"CNb

k6gC=A("@9C%+.+9-1$'2"C

τ1(G)U

*g, . $ &1$#$ $ ". 19 6øC/>"!$'@7 &IC-02"-0/.-b,N0C%&)C%2:/á!/>!_&)$'2:/>@9!_C%2N1$,M'-1XB"C"B("@.C&I-0C%@e$'@9"@.Cc5(H$'B"@DN0C%+eB"25-1$'2"+"C+9/[email protected]"#=/>B"@9C%+yC/yN1C+_("@.$a"B"-7/>+_"C +/>@.B5#/>[email protected]%+c(",@FECdÁC%@9& 2uC/ á,B"M':/IP]F °TôU B"-0+_N1C4@9!%+.B5N0/wÜ/(H$'B"@áN1C%+GB"2"-0$'2"+á"CK+/>@.B5#/>[email protected]%+e!/.!K!/.C%2"5BGoNbN1$,M'-1XB"C&)$'2E,"-0XB"C5B`+.C#%$'2" $,@."@9CP],B+x5c

tA"CUvÜTVUZ2"C("@9C%B5h,C #$'2"+/>@.B5#/>-7h'Cy(*CB5/?/[email protected]_/>@9$'B5h'!C ",2"+yPQJLKaR,,TVU"$# ÌXZ¾%aÉkÈ ;;( &($, 8?K !^+D6Z;/@CI

τKX1 b?/Ha7:MOQKXB :?@C[DsD]IL/@CI

AD]IA′ FHDNL

τR

L]I[]K8a]IKX[DNLQc-;/@CIq ∈ N

D]Iϕ ∈ MSOLq(τ)

c BE/[L @CB-D@? @EL]IDl ∈ N

_KX13D_<U/[a>AKHBEDO]//BEND]1+13DB : 0, 12l → 0, 1

D]I\+/KH[ I/KHIi ∈ 1, . . . , l

ϕi, ϕ

′i ∈ MSOLq(τ)

ID]BL AQKD

A⊔A′ |= ϕ

L]@yD]I LD]KHBED]>D]1+IL]@B(a1, . . . , al, a

′1, . . . , a

′l) = 1

/ai = 1

L]@$D]I LD]KHBED]>D]1+I L]@A |= ϕi

D]Ia′i = 1

L]@ D]IiLD]KHBED]>D]1+I L]@A′ |= ϕ′

i

\+/KH[PI/KHIi ∈ 1, . . . , l

c2 [\,B5/[email protected]%+å/>C@.&)C%+c:N1Ce/>5!%$'@ %&)CaU\y"-0/åXB"C

Thq(A⊔A′)

C%+/áB"2"-1XB"C&IC2:/"!=/>C@.&)-12"!D(E@Thq(A)

C/Thq(A′)

U

MSOL

/+ #* . &($'68 92 684<,/9øCK+.#w5!%&I"C/>@>2"+.N1/>-0$'2 +>,2"+GXBE,2:/>-7OE#/.C%B"@GC%+/eB"2`#%,+á(E,@9/.-1#B"N1-0C%@"B`+9#w"!%&I"C/>@.,2"+ö6NbÜ/>-1$,2`&I$,2E,"-0aB5C_"BL+9C%#$'2"^$,@."@9C PlDJHG?R'R"c8åF ÜTôUtA$'-7/

τ = R1, . . . , RkB"2uh,$A#%,3"B"N1,-1@9CC/(H$'B"@

i ∈ 1, . . . , kcº+9$'-0/

ai

Nô[\,@.-7/>!y"CRi

Uø?2ÎÍ # ÌaÉkÀkËÈLm¾°ÀEÎ%ÂfÀ5mfH Î%ÀEÎ 3pÀEH° Í'À5mÈHp¾4E,2"+ τ C%+9/ Φ = 〈ψ1, . . . , ψk〉$(H$'B"@D/>$'BA/

i ∈ 1, . . . , kcψi ∈ MSOL0(τ)

Á#,[]C%+9/6oÜ6g"-1@9CC%+/B"2"CdÁ$'@9&B"N0C&)$'2E,"-0XB"C"Buh,$A#%,3"B"N1,-1@9Cτ+>2"+XBE2X/.-0OE#%/>CB"@ cH

ai

h,@.-1,3"N1C+DN1-03"@.C+D"B^("@9C%&)-1C@D$'@."@9C,cZC/?2[\ (E,+?"C_h,@.-b3"N1C_N0-13"@9C_"B+.C#%$'25^$'@9"@.CUtA$'-7/

AB"2"C

τ6Y+/>@9B"#/.B"@.CU KN0$'@.+

Φ∗(A)C%+9/`Nb

τ6Y+/>@.B5#/>[email protected]"CÆ5$'& -12"C

AC/`(*$'B5@I/>$,B5/

i ∈ 1, . . . , kcR

Φ∗(A)i = (x1, . . . , xai

) : x1, . . . , xai∈ A

C/A |= ψi(x1, . . . , xk)

cá$eai"!+.-1M,2"CNô[\,@.-7/>!"C

Ri

UtA$'-7/

ϕB"2"CdÁ$'@.&B"N1C+.B5@N0Cyh,$A#%,3"B"N1,-1@9C

τUmKN0$'@.+

Φ](ϕ)C%+/KN1 dÁ$'@9&B"N0C

ϕE,25+KNbaB5C%N1N0C_$'2 @9C%&)("Nb,#!N1C+K$a#%#B"@.@9C%2"#C%+"C

Ri(x1, . . . , xa)(E@

ψi(x1, . . . , xa)UN1B"+KdÁ$,@.&)C%N1N0C%&)C%2:/c

Φ](ϕ)(HC%B5/?/[email protected]"!=OE2"-1C@9!%#%B5@.+.-7h'C&IC2X/ §Φ](∀X,ϕ′) , ∀X,Φ](ϕ)

C/Φ](∃X,ϕ′) , ∃X,Φ](ϕ)

U§Φ](ϕ1 ∧ ϕ2) , Φ](ϕ1) ∧ Φ](ϕ2)

C=/Φ](ϕ1 ∨ ϕ2) , Φ](ϕ1) ∨ Φ](ϕ2)

U§Φ](¬ϕ′) , ¬Φ](ϕ′)

U§Φ](Ri(x1, . . . , xai

)) , ψi(x1, . . . , xai)(*$,B"@D/>$'B5/

i ∈ 1, . . . , kU§

Φ](U(x)) , U(x)+.-UC%+9/B"2"C_h,@9-b,3"N0CK"B+.C#%$'25^$'@9"@.CU§

Φ](x = y) , x = yUÃ242"$,/.CeXB"C+.-

ΦC%+/åB"24+.#w5!%&I5C/>@.,2"+.N1/>-0$'2+.,2"+XBE,2:/>-7OE#Ü/>C%B5@%c'N1$'@9+

qr(Φ](ϕ)) = qr(ϕ)U

"$# ÌXZ¾%aÉkÈ ;;(;;(y, 6_;/@CIτKX1b?/Ha7:MOQKHB :?@C[D

AKH13D

τRL]I[aKEa]IKX[D

ΦKX1 LHaN^+]>g:FHD I*[:?1`L]R

B :?I@/1LQ:?1`L AQK!:?1+I@ a7:?ID]KX[$D]IϕKH13D <U/[]>AKHBED L]KH[ BEDib?/Ha7:MOQKXB :?@C[D

τc BE/[L

A |= Φ](ϕ) ⇐⇒ Φ∗(A) |= ϕ.

6847684<,/9MSOL

2/( <$^2 2>688#+M6"(%$g2 &1$ $ ( &1$ + 4#$ !1,e9*^$"$# ÌXZ¾%aÉkÈ ;;( ;($;/@CI KH1 D]1+I*@D][

kiKX1 b?/Ha7:MOQKHB :?@C[ND

τ kAD]I

ϕ ∈ MSOL(τ, ∅, X) ?XNLQc(=aB

D@? @EL]ID KX1 :?B YH/[a@CIC^H>D \+/B `13/>A@:?BX\+/KX[ BED \m[/?OQB]>D L]KH@Cb:?1+I 1+I*[ND 13D

τkRL]I*[aK8a]I*KH[D

AFHD B :?[*YHD]KX[ FHDja]B @BAQK+D$:?K9\mB KML

kD]IKH13D

kR7D@?=\m[DNLNL]@/1g\+/KH[

A;/[aI*@D C ⊆ 1, . . . , |V |

ID]B>AQK+Dr ∈ C

L]@3D]ImLD]KHBED]>D]1+IL]@+@CB`D@? @EL]ID KX13D\`:?[aI@CI*@/1A1, . . . , ArFHD

AID]BCBED AQK+D

∀i ∈ 1, . . . , r, 〈A, Ai〉 |= ϕ(Ai).

]>/1`L]I[:?I*@/1mctA$'-7/q = qr(ϕ)

U[\-0"!%Cy"CyNb4("@.CB5h'CC%+9/"Cy#,N0#%B"N0C%@@.!#%B"@9+.-7h'C%&)C%2://>$'B"+?N1C+&B"N0/.-76gC%2"+9C%&3"N0C%+D"C+/."!%$'@9-1C+"C_/>$,B5/>C+?+/>@9B"#/.B"@.C+K,(5(E,@>-1+.+.,2:/?E,2"+Nb5!%#%$,&I(H$'+.-7/>-0$'2 "CNb+/>@.B5#/>[email protected]

N1B"+DdÁ$'@9&ICN1N1C&IC2:/ca$'2L#%,N1#B"N1CK@.!%#B"@.+9-0h,C%&)C%2:/K+9B"@Nô[\,@[email protected]_"C"!#%$'&)(*$,+.-0/.-1$'2 NV[\C2"+.C&3"N0CS(A) =

〈Thq (〈A, Ai〉)〉i∈1,...,r : A1, . . . , Ar

C%+/B52"C (",@9/.-0/>-0$'2`"CA.

, 5 ?3

[\C2"+.C&3"N1CS(H$'B"@DN1C+D+9/[email protected]"#=/>B"@9C%+

·iC%+/N0C_+.B"-7h,2:/

S(·i) =〈Thq (〈A, v〉),Thq (〈A, ∅〉), . . . ,Thq (〈A, ∅〉)〉

#@)-0Ne2[ aBº[\B"25CL+9C%B"N0C(",@9/.-0/>-0$'2 (H$'+.+9-13"N0C "Bý5$'& -12"C`5C·iUøC%+ />5!%$'@9-1C%+ (HC%B5h,C%2:/I/>@9C#N1#%B5N1!%C+GC2I/.C%&)("+G#$'2"+/w,2:/cX#%,@GN0C%+á+9/[email protected]"#=/>B"@9C%++9$'2:/e"C/w-1N1N0C%+<OA5!C%+eC=/-0NR)yB52 2"$'&3"@.CO5A!"C_dÁ$'@9&B"N1C+o/>C%+/>C@%U

$'B5@`/>$'B5/R ∈ τ

C/i1, . . . , ia ∈ 1, . . . , k

-1NC=A-1+/>CuB"2 +9#w"!%&I "CÆ/>@.,2"+.N1/>-0$'2 +.,2"+XBE,2:/>-7OE#Ü/>C%B5@L(H$'B"@ηR

(i1,...,ia)

ΦηR

(i1 ,...,ia)= 〈ψR′ : R′ ∈ τ k〉

$eψR′ = R′(x1, . . . xa′)

+9-R′ ∈ τ k\R

Á$a′C%+/Nô[\,@.-7/>!á"C

R′ =cÜC/ ψR = R(x1, . . . xa)∨(Pi1(x1)∧Pi2(x2) . . .∧Pia(xa))U

KC &I&ICc(H$'B"@/>$,B5/i, j ∈ 1, . . . , k

c-0N<C=A-0+9/>C B"2Ä+.#w5!%&I5C /.@>,25+.NbÜ/>-1$,2Æ+>,25+ XBE,2:/.-76OE#%/>CB"@(H$'B"@ρi→j

cXB"-áC%+/N1C)+.B5-0h,2:/Φρi,j

= 〈ψR′ : R ∈ τ k〉$

ψR′ = R′(x1, . . . xa′)+9-

R′ ∈ τ k \ Pi, Pjf$

a′C%+9/?NV[Q,@9-0/.!"C

R′ c ψPi= Faux

C/ψPj

= Pi(x1) ∨ Pj(x1)U

tA$'-7/fB"2"C4$'(H!%@./>-0$'2

ρi→j

$'BηR

(i1,...,ia)

cC/Φ+9$'2 +.#w"!& "C/.@>,2"+9Nb/.-1$'2L#%$'@[email protected]+.(H$'2"E,2:/%U

KN0$'@.+_S(f(A)) = 〈Φ](Ti)〉i∈1,...,r : 〈Ti〉i∈1,...,r ∈ S(A)h'C#

Φ](T ) = Φ](ϕ) : ϕ ∈ TU

l$'&)&ICN1C%+&B"N0/.-76gC%2"+9C%&3"N0C%+G,("(E@9/>-0C%2"25C%2:/oB"2B"2"-7h'C@.+D[\,B ("N0B"+(r+ 1)|MSOLq(τ,∅,X)|!%N0!%&)C%2:/>+c

S(f(A))(HC%B5/?=/>@.C#%,N1#B"N1!_C2/.C%&)("+

(r + 1)|MSOLq(τ,∅,X)|+1 U$'B5@KNô[]$'(H!%@./>-0$'2

⊕cH$,2^B5/.-1N1-0+.C_N0C/."!%$,@7 %&)Cy"CFEC=dÁC%@.&I,2A6 á,B"M':/%UmtA$'-0/

AC/A′ "CBA τk

6+9/[email protected]"#=/>B"@9C%+cºC/K+.$'-7/A1 ⊆ A

C=/A′

1 ⊆ A′ U*øC /."!%$'@ %&)C2"$,B"+K"-7/KXB[]-1NC+9/_(H$'+.+9-13"N0C 5C#N1#%B5N1C%@Thq(〈A

⊔A′, A1 ∪ A

′1〉)"C("B"-1+

Thq(〈A, A1〉)C=/

Thq(〈A′, A′1〉)U+KC("N1B5+%c"#$'&I&)C_Nb/>,-1N0N1C?"C#%C+C%2"+9C%&3"N0C%+C%+/3*$,@.2"!C4(E,@_B"2"C4#$'2"+/w,2:/>C4O5A!C Á"!%(HC%25E,2:/>C)"C

τkC=/"C

q=c#C%Nb`+9Cdf,-0/C%2/>C&I(5+?#$'2"+/w,2:/U

tA$'-7/Ma ∈ S(A)

C=/Mb ∈ S(A′)

UlE,XB"CáC%2"+9C%&35N1CTa

"CMa

#%$'@[email protected]+.(H$'2" oThq(〈A, A1〉)(H$'B"@B"2

A1 ⊆ AC/<#wE,XB"CC2"+.C&3"N0C

Tb

"CMb

#%$'@[email protected]+.(H$'2"AoThq(〈A′, A′

1〉)(*$,B"@B52

A′1 ⊆ A′ UÃ2L(HC%BA/"$'2"##N1#%B5N1C%@

Thq(〈A⊔A′, A1 ∪ A′

1〉)cEXB"CNô[]$'22"$,/>C@>

Ta Tb

UÃ2 #%$'&)&IC2"#%C h'C%# NV[\C2"+.C&3"N0CD = 〈〉 × S(A) × S(A′)

c<(5B"-1+ $'2 !/.C%2"D(",@/>$,B"+N1C+4/>@.-0("N1C=/>+4XB"-D(HC%B5h,C%2:/`=/>@9C$,35/>C2aB5+ []B"2/>@.-0("N1C=/

(M,Ma,Mb) ∈ DC%2Ï+.B"("([email protected]&I,2:/)B"2C%25+.C%&3"N1C

Ta

"CMa

C/B"2ÆC2"+.C&3"N1CTb

"CMb

cZC/C%2Æ@.°89$'B5/>,2:/Ta Tb

oMUHÃ2Lh,$'-0/XB"CXBE,2"

D2"C(*$,B"@.@. (5N1B"+/[email protected]!/.C%2"5BcZ$'2,B"@>

S(A⊕A′) = M : (M, 〈〉, 〈〉) ∈ DUøC%+$'(H!%@>Ü/>-1$,2"+

+9C dÁ$,2X/C2Ä/.C%&)("+_#%$,2"+9/>,2:/UWYN^ ,B ("N0B"+

n2 df%$'25+_"C #%$'&35-12"C@_B"2!%N0!%&)C%2:/"CM1

mh,C%#DB"2 !N1!%&)C%2:/<"CM2

UMl$,&I&)CMcM1

C/M2

("(E,@/>-1C2"2"C2X/oB"24B"2"-7h'C%@9+[\,B^(5N1B"+(r + 1)|MSOLq(τ,∅,X)| !N1!&IC2X/.+%c D I,BL("N1B5+ (r + 1)3×|MSOLq(τ,∅,X)| !%N1!&IC2:/>+%cZC=/(HC%B5/?"$'25#y+9C#,N0#%B"N0C%@DC%2/>C&I(5+

(r + 1)3×|MSOLq(τ,∅,X)|+2

øC/."!%$,@7 %&)CAUux)("("N1-0XB"!_+.B"@N1+9/>@9B"#/.B"@.Cτ1[]B"2M'@>(""C2"$'B"+"$,2"2"C"-1@9C%#=/>C%&)C%2:/i

: Z¾ÜHÂfÂfÀ"f¾°È ;;( ( /KHIW\m[/?OQB]>DFHD\`:?[aI*@CI@/1+13D]>D]1+IMSOL1

\+D]KXI]I*[D [NL/B K0D]1qID]>\+L\+/RB `13/>A@:?B L]KH[$BEDNL_Y?[:]\+^+DNLFHD$B :?[*YHD]KH[AFHD a]B @BAQK+D9O]/[a13ND L]@ B :

kR7D@?=\m[NDNLNL]@/1ZDNL]IFH/1+13NDA:?b?Dba BED

Y?[:]\+^+DQc

`ÈaÉkÀ"¾B;pÈ(#=]Bm13V DNL]I \`:L13baNDNLNLQ:?@C[ND FHD a7:?Ba]KXBED][Thq(A)

c_=]B!L]KsI FHD I[/KXb?D][Ψ ⊆ MSOLq(τ)ID]B AQKD\+/KH[ I/KHI

ψ ∈ Ψ \+/KH[ I/KHILHaN^+]>g:FHD I*[:?1`L]B :?I@/1

Φ@aN/[a[DNL\+/1mFM:?1+I:?K"? /N\+][:?I@/1`L

ρD]Iη

Φ](ψ) ∈ ΨD]I \+/KH[ I/KXIDNL$BEDNL <U/[]>AKHBEDNL

ψi

:]\M\`:?[:?@ELNLQ:?1+IyFM:?1`LiBED I^+N/[]>D$FHD D<UD][a>g:?1+R +:?K?Y^HI;\+/KX[B :<U/[a>AKHBED

ψψi ∈ Ψ

cD]I D]1`LD]>gOQBED9FHDy<U/[a>AKXBEDNL \+D]KHI ]I*[D a7:?Ba]KHBED]1 aN/>A>D]1:?1+I \`:?[B7V D]1`LD]>gOQBED

ϕ/

ϕDNL]I

B :\m[N/N\m[a@]IMSOL

Ab?][a@ D][ D]IyD]1:Y?[:?1mF?@ELNLQ:?1+I I:?1+I#AQKD \+/LNL]@OQBED-B7V D]1`LD]>gOQBEDD]1 :SU/KHI:?1+IΦ](ψ)

D]I BEDNLψi

\+/KX[iI/KXIψ ∈ Ψ

c K`:?1mFB7V D]1`LD]>gOQBED 13D\+D]KHIW\mB KML]I[D:Y?[7:?1mF?@ :?BE/[NLi@CB :B :\m[/N\m[a@]I FHD]>g:?1mFHNDQc

0 0 B4+ 684<,/9 3$ ( W!1,%$g2R+-$39 6$

«[email protected]'CB"@ "C^#N1-1XB"C"CLNb +9/>@9B"#/.B"@.Cτ2(G)

C+9/`!M:,N0C oÄNb Nb,@9M'C%B"@ "CL#%N0-1XB"C"CI(G)

c#@τ2(G)

C+9/yNbI+9/>@9B"#/.B"@.Cτ1(G)

cøB"M'&)C%2:/>!C)º[\B"25C)@9C%N1/>-0$'2ÆB"[email protected]

ÁaB5-2"C #wE,2"M'C (E+NbÆN1,@.M,C%B"@5C#N1-0aB5C =UK$'2"#+.-DN1ÆNb,@9M'C%B5@4,@.3H$'@9C%+.#C%2:/>C"CGC+9/)3H$'@.2"!C (E,@

kcüN1ÆNb,@9M'C%B5@,@93*$'@9C%+9#%C%2:/.C"C

I(G)C%+9/<,B ("N1B"+

kP].G?R,,TVcmC/NbDN1,@.M'CB"@"Cá#%N1-0XB"CåC%+9/<,By("N1B5+

3×2k−1 Pul GRÜTôU: Z¾ÜHÂfÂfÀ"f¾°È ;;(";( m/KXI+\m[/?OQB]>DFHD\`:?[aI*@CI@/1+13D]>D]1+I

MSOL2

\+D]KHI ]I[D-[NNL/B K D]1GID]>\+L\+/B `R13/>A@:?BmL]KH[ BEDNL-Y?[:]\+^+DNLiFHD B :?[*YHD]KH[i:?[=O]/[NDNLHaND]1+ID O]/[a13ND L]@ B :GFHbaN/>\+/L]@CI*@/1DNL]IFH/1+13ND :?b?DbaBEDY?[:]\+^+DQc

,2"+<#%Ce"[email protected]"-0C%@#,+c+9-5Nb?"!%#$'&)(*$'+9-0/.-1$'2y2[]C%+9/<(E,+ü5$'2"2"!CDh'C%#N1CM,@>,("5C,cÜ$'2(HC%B5/<B5/>-0N1-1+9C%@Nô[\,N0M'$'@.-7/>"&)C?"$'2"2"!E2"+yP $a x°T(*$,B"@DC%2/>@9$'B5h'C@B52"CC%2/>C%&)("+N1-02"[email protected]

kjöi Gf>o*jyxoft vÃ2Ih'$,-0/ádf,#%-0N1C%&)C%2:/åXB"CNô[]$'2I(HC%BA/@9!%+.$,B""@.C"-1@9C%#=/>C%&)C%2:/GN1C("@9$'3"NE &IC"C?#%$,N1$'@./>-0$'2 +.B"@áN1C+M'@.,(""C+?5CyN1,@.M,C%B"@D"C#N1-1XB"CB(5N1B"+

kC%2/>C+9/>,2:/KN0C%+D("@9$'("@.-0!/.!%++.B"-7h,2:/>C%+

§pC+9/Nb("@9$'("@.-0!/.!

XC+9/?B"2C%2"+9C%&3"N0C+9/w3"N1C Hc§ (H$'B"@e/.$'B5/

i ∈ 1, . . . , kcpi

C%+/N1("@.$,("@.-0!/>! X(H$'+.+ %"CyB"2+.$'&)&IC=/D!/.-1XB"C=/>!

i*U

WYNº+9B /D"$'25#"C1 + k

(5@.$'("@9-1!=/>!%+UEÃ2`h'$,-0/df#%-1N0C%&)C%2:/GaB5C_#%C%+("@.$'([email protected]!=/>!+e+.$'2:/D5C%+D("@9$'("@.-0!/.!%+MSOL1 p

(*CB5/?+%[]!%#@.-1@9C∀u, v ∈ X,¬Adj(u, v)

c"C/pi

(*CB5/?+%[]!%#@.-1@9C∃v ∈ X,Pi(v)

UN1-0+9/>C?"C+e("@.$,("@.-0!/>!+ o /.C%+/>C%@M'!%2"!@.!C+.B"-7h,2:/[email protected]& ,@9XB"CK"CKNbyO"2`"CKN1 +9C%#/.-1$'2 AUQSAU\"c(H$'B"@k = 2

C/ϕ = ∀u, v(¬Adj(u, v) ∨ u 6∈ X ∨ v 6∈ X)

f#[\C+9/96oÜ[email protected] XC%+/_B"2uC%25+.C%&3"N1C+9/>,3"N0C =c5C%+9/Nb+9B"-0h,2:/.Cg

, 5 ?3

ϕ1 = ∀u, ∀v, (¬Adj(u, v) ∨ u 6∈ X ∨ v 6∈ X)

ϕ2 = ∀u, ∀v, (¬(Adj(u, v) ∨ (u ∈ P1 ∧ v ∈ P2) ∨ (v ∈ P1 ∧ u ∈ P2)) ∨ u 6∈ X ∨ v 6∈ X))

ϕ3 = ∀u, ∀v, (u 6∈ P1 ∨ v 6∈ P2 ∨ u 6∈ X ∨ v 6∈ X)

ϕ4 = ∀u, ∀v, (v 6∈ P1 ∨ u 6∈ P2 ∨ u 6∈ X ∨ v 6∈ X)

ϕ5 = ∀u, (u 6∈ P1 ∨ u 6∈ X)

ϕ6 = ∀u, (u 6∈ P2 ∨ u 6∈ X)

ϕ7 = ∀v, (¬(v ∈ P1 ∨ v ∈ P2) ∨ v 6∈ X)

Ã2L@9C%&I,@.XB"CXB"Cϕ1 ≡ ϕ ≡ p

cEXB"Cϕ5 ≡ ¬p1

C=/XB"Cϕ6 ≡ ¬p2

U$'B5@áN1_("@.$'([email protected]!=/>! eC%25+.C%&3"N1C+9/>,3"N0C *c:(*$,B"@

k = 2c'N1_&I!=/>"$a"CDM'!2! %@9C?5$'2"#

7(5@.$'("@9-1!=/>!%+U

$,B"@k = 3

cÜCN1N0CGM'!2! %@9C27("@.$'([email protected]!=/>!+%cÜC/ü(H$'B"@

k = 4cÜC%N0N1CåM'!%2` %@.C

129("@.$,("@.-0!/>!+%Ueå2("@>/.-1XB"Cc$'2 $'35/>-0C%2:/"$'2"#G3*C%,B"#%$,B"(&)-1CBAyXB"CN1?3H$'@.25C+.B5(*[email protected]"@.Cå/>5!%$'@9-1XB"C5C|MSOLq(τ1,k ∪X)|fXB"-øC+9/?B"2"C/>$,B"@D[]C=A(*$,2"C%2:/>-0C%N0N1C_C%2Nb4("@9$,dÁ$'2"5C%B"@D"CXBE,2:/.-0OE#%/>-0$'2 =UÃ2Æ(*CB5/@9C%&I,@.XB"C@XB"CNô[]C%2"+9C%&3"N0C 5C%+K(5@.$'("@9-1!=/>!%+M'!%25!%@.! (*$,B"@N0C (5@.$'3"N %&)C @ +39')>4;)>4 2

. "$#0%, ++ .) % +@9 2 @9.4 ) ϕC+9/)C=5,#=/>C&IC2X/)NV[\C2"+.C&3"N1C 5C%+)("@.$'([email protected]!=/>!+4(*$'B5@/>@9$'B5h,C%@KN0C("N1B"+?M'@.,2"^C2"+.C&3"N0C :2X/KNb ("@.$,("@.-0!/>!

ϕcZ(H$'B"@Nô[\,N1M,$'@.-7/>"&)C"CIPlDJ GR'RÜTôU $,B"@N1C ("@9$'3"N %&)C /.!"$#&%' *),+ #&%'c+.B"@yN1C+M'@>,(5"C%+y"C)(*C=/>-0/.C Nb,@9M'C%B"@_5CI#N1-1XB"CcøNô[\,N1M,$'@.-7/>"&)C"C)PulDJHGR,RT+.C&3"N1C_5$'2"#B5/>-0N1-0+>,3"N0CKC2L("@./>-0XB"C,U

kj tuvjAfwzoVt vÃ2 u&)$'2:/>@.! XB[]B"2"C2"$'BAh'C%N0N1C#%N1,+.+9C"C("@.$'35NE %&)C%+4C+9/)(H$'N<a2"$'&)-bN1C)+.B"@)N1C+4#%Nb+.+.C+ "[email protected]'CB"@D"C#%N0-1XB"C3*$,@.2"!C,cE#%CyXB"-ø@>m89$'B5/>C"C+(5@.$'3"N %&)C%+o4Nb4"!Y87o)M'@.,2""CN0-1+9/.C_"C("@.$,3"NE &IC+(H$'N<a2"$'&)-bBA (H$'B"@N1C%+?M'@.,(""C+K"C Nb,@9M'C%B"@?"C #%N1-0XB"C 3H$'@92"!%CUåBÆN1)@9C%N1/>-0$'2LC2:/>@.CNb Nb,@9M'C%B"@,@93*$'@9C%+9#%C%2:/.C^C/INb [email protected]'CB"@ "CL#N1-0aB5C5B M'@>(""C[\-02"#%-0"C%25#%C,cå$'2 ÄB52"C^2"$,B5h'CN1N1C#Nb,+9+.C"C("@9$'3"NE &IC+D(*$,N<a2"$'&)-b,Ba`(H$'B"@DN0C%+DM'@.,(""C+?5CyN1,@.M,C%B"@,@93*$,@.C%+9#%C2X/.C 3H$'@92"!%CU;!%,2"&)$'-12"+cG$,2 2"C^#$'2"2E 1/`C2"#%$,@.C^(",+`NbÄ#%$'&)("N0C=A-0/.!L"CL#%C@9/w-12"+I("@.$'35NE %&)C%+I2E/.B"@.CN1+%c"$'2:/ $'2Ä(*CB5/#-0/.C%@ 45 1 68< 9 2 "J+-)>4 :. Á("@.$'35NE %&)C4$«Nô[]$'2uh'C%BA/#%$,N1$'@9-1C%@_N0C%+y,@9/>C+h,C%#B"22"$'&[email protected]_&)-12"-0&B"& "C#%$'B"N0C%B"@9+ UÃ2«2"C)+>-0/2"$'2«("N1B5+y(",++.-åNb [email protected]'CB"@_"C #%N0-1XB"C (*CB5/ ,-15C%@(*$,B"@_"!%#-1"C@ C2«/.C%&)("+(*$6N<a2"$,&I-1,N<+9-áB52k#%C@9/>,-12Ä(E,@.,& =/>@9C ÁNb,@9M'C%B5@y5CI#N1-1XB"CcøNb,@9M'C%B5@ ;KWlKU0U7U yC%+/3H$'@92"!I(",@B"2

kU

l$'&35-12"!h'C%#_N0C%+,N1M'$,@.-0/."&)C%+(H$'B"@N1yN1,@.M,C%B"@á"C@.,2"M`P\Ã t5R xTVca#%C%N1 2"$,B"+e"$'252"C%@.,-0/GB"2 ,N0M'[email protected]/>"&)C(H$'N7A25$'&I-1,N(*$,B"@"!#%-15C%@_+.-<Nb N1,@.M'CB"@K5C4#%N1-0XB"CC%+/3*$,@.2"!C4(E,@kckO5A!,Uø?2Ä,B5/>@9C("@9$'3"NE &IC_$,B5h'C@9/?-12:/>[email protected]+.+>2X/C%+/?N1C_("@9$'3"NE &IC 45 2 " 2 9.@<.45!"$ 1 AC98+3@<.U

M

kj vv4 #, 4$ . ,9 #&/4 $ & 2 $+-,/9#& ,%&/%$

tA$'-7/τB"24h'$a#,35B"Nb,[email protected],øC%+ gZ¾ÜÉqpÂÁÈaÎÀ5*Ék;pÈaÎe+.$'2:/åN0C%+üdÁ$'@9&B"N1C+y

VraicFaux

cx1 = x2C/

R(x1, . . . , xa)cD$

x1, . . . , xa

+.$,2X/"C+h,@9-b,35N1C%+ceC/RC%+9/LB"2+@a&3H$'N1Cu5C«@9C%NbÜ/>-1$,2 "Bh'$a#3"B"[email protected]øC%+ gZ¾ÜÉqpÂÁÈaÎË pýÇ¾°ÈaÉ ÁÈa¾4Z¾ÜË¾ÜÈ)+.$,2X/N0C%+dÁ$,@.&B"N0C%+e#$'2"+/>@.B5-0/>C+D@.!#%B"@9+.-0h,C%&)C%2:/C%2B5/.-1N0-76+>2X/N0C%+edÁ$'@9&B"N0C%+D/.$'&)-1XB"C%+c5C=/N1C%+D#$'2"2"C#/.C%B"@9+

∨f$'B c

∧fC/ =c

¬f2"$,2 =c

∀f(H$'B"@G/>$'B5/ C/

∃f-0NC=A-0+9/>C =U§ tA-ϕC+9/?B"2"CdÁ$'@.&B"N1C_Ü/>$'&)-1XB"Cc",N0$'@.+

ϕC+9/?B"2"CdÁ$'@.&B"N1C5BL("@9C%&)-1C@e$'@[email protected],c§ tA-

ϕC=/ψ+.$,2X/)"C+4dÁ$'@.&B5N1C%+"Bý([email protected]%&)-1C@$'@9"@.Cc,N1$,@.+

ϕ ∨ ψcϕ ∧ ψ

C=/¬ϕ

+.$'2:/)"C%+dÁ$'@9&B"N0C%+D"B("@9C%&)-1C%@D$,@."@9C,c§ tA-ϕC+9/DB"25CdÁ$'@9&B"N1C?"B`([email protected]%&)-1C@G$,@."@9CKC/

xB"2"Ch,@9-b,35N1C,ca,N0$'@.+

∀x, ϕC/∃x, ϕ

+.$'2:/D5C%+dÁ$'@9&B"N0C%+D"B("@9C%&)-1C%@D$,@."@9C,U2^ÂÁz* ;pÈ^ËWp Î%ÈaÍ'HË Z¾ÜË¾ÜÈZcEN1aB",2:/>-0O"#/.-1$'2 C+9/(*$'+9+.-03"N1C+9B"@N1C+?@9C%N1/>-0$'2"+%U5øC%+DdÁ$'@6&B"N0C%+DÜ/>$'&)-1XB"C+(HC%B5h,C%2:/?"$'2"#_B"+.+9-/[email protected]"C_N1 dÁ$'@9&IC

X(x1, . . . , xa)c5$

XC+9/B"25Ch,@.-b3"N1C"BL+.C%#$'2"L$'@."@9C,c

a+9$'2Æ@.-0/.!C/

x1, . . . xa

"C%+?h,@.-b3"N1C+D"BL([email protected]%&)-1C@$'@."@9C,UE2ÆÂÁzH;pÈÆÉ«EÀEË ;pÈ ËWp Î%ÈaÍ'*Ë Z¾Ë¾°ÈZc<NbLaB",2:/>-0O"#/.-1$'2 2[]C%+/)(*$'+9+.-03"N1CIaB5C+9B"@4N1C+h,@.-1,3"N0C%+C=/ [email protected]/.-1$'25+eB"2E,[email protected]+ ÁXB"-ø(*CB5h'C2:/K=/>@.C_haB"C+#$'&I&)C_"C+?+9$'B"+ö6YC%25+.C%&3"N1C+"B"$'&I,-02"C U W4(! $g2 B4 !# $^2[\C2"+.C&3"N1C"C+?h,@9-b,35N1C%[email protected]%+D(HC%BA/=/>@9Cy#%,N1#B"N1!K@.!%#B"@.+9-0h,C%&)C%2:/$§ +9-

ϕC%+/åB"2"CedÁ$'@9&B"N0C/.$'&)-1XB"C,c,N0$'@.+

libre(ϕ)C%+/åNô[]C%2"+9C%&35N1Ce"C+üh,@.-1,3"N1C+<("(E,@.,-1+9+>,2:/E2"+

ϕc§

libre(ϕ ∧ ψ) , libre(ϕ) ∪ libre(ψ)c§

libre(ϕ ∨ ψ) , libre(ϕ) ∪ libre(ψ)c§

libre(¬ϕ) , libre(ϕ)c§

libre(∀X,ϕ) , libre(ϕ) \ XcA$

XC%+/B"2"CKh,@9-b,3"N0C"B ("@.C&I-0C%@á$'B`"B +9C%#$'2"$'@[email protected],c§

libre(∃X,ϕ) , libre(ϕ) \ XU

,/& <$^2)@9C%NbÜ/>-1$,2

|=C%+/?"!OE25-1C"CNbdf$'2+.B"-7h,2:/>C9§

〈A, z〉 |= ϕ = R(x1, . . . , xr)+9-

(z′(x1), . . . , z′(xr)) ∈ z′(R)

cø$ez′C%+9/ Nb dÁ$'2"#=/>-0$'2 aB5-!=/>C%25

z,BAÆ+@a&3H$'N1C+y5C

τf#,[]C%+/96oÜ6Y"[email protected]

z′(R) = RA (H$'B"@/.$'B5/[email protected]%N1/>-0$'2 R "CτcøC=/

z′(X) = z(X)(H$'B"@e/.$'B5/.Ch,@9-b,35N1CK",2"+N1C_"$,& ,-02"C_"C

z U§

〈A, z〉 |= ϕ1 ∧ ϕ2

+.-〈A, z〉 |= ϕ1

C=/〈A, z〉 |= ϕ2

c§〈A, z〉 |= ϕ1 ∨ ϕ2

+.-〈A, z〉 |= ϕ1

$,B〈A, z〉 |= ϕ2

c§〈A, z〉 |= ¬ϕ1

+9-ø$'22[Q (E,+〈A, z〉 |= ϕ1

c§〈A, z〉 |= ∃X,ϕ1

+[\-0N5C=A-0+9/>CC ⊆ A

C ∈ A

+9-XC%+/á"B ("@.C&I-0C%@ü$,@."@9C =c

〈A, z[ CX

]〉 |= ϕ1c

, 5 ?3

§〈A, z〉 |= ∀X,ϕ1

+9-(H$'B"@G/.$'B5/C ⊆ A

C ∈ A

+.-XC%+9/e"B("@.C&I-0C%@á$'@[email protected] c

〈A, z[ CX

]〉 |=ϕ1

UtA-〈A, z〉 |= ϕ

c$'2Ä"-0/aB5C〈A, z〉

C%+/yB"2kÉ«øËW%aÂÁÈL5CϕUWtA-üN1C45$'& -12"C5C

zC%+/y2aB5Nôc$'2!%#@.-0@>

A |= ϕU

14 <$39#+-$)&($^2 %,". $g2Ã2I"-7/åXB"C"C%BA dÁ$,@.&B"N0C%+

ϕC/ψ+9$'2:/áÌ,3pr¿"ÀEÂÁÈa*mÈaÎ,c'XB"CNV[\$'2 2"$/>C%@.(E,@

ϕ ≡ ψc'+9-E(*$,B"@/>$,B5/>CI+9/>@9B"#/.B"@.C

AC/,+.+9-1M'2"/>-0$'2

zc〈A, z〉 |= ϕ

+.-áC/+.C%B5N1C%&)C%2:/ +.-〈A, z〉 |= ψ

U tA-á$'2ÄO5ACq ∈ N

cÜ-1N KB"2 2"$,&3"@.C2"$'2OE25-""CdÁ$'@9&B"N0C%+<5CD("@.$,dÁ$,2""C%B5@ü"CeXBE,2:/.-0OE#%/>-0$'24,B4("N0B"+qcÜ& ,-0+-1N2º[«LXB[]B"2 2"$,&3"@.C OE2"-á"C dÁ$'@.&B"N1C+ 2"$,2k!%XB"-7h,N1C2X/.C%+UüJ^N1"CB"@.CB"+.C&IC2X/%c<NV[\!aB5-0h,N1C2"#%CC%2:/[email protected]%+edÁ$,@.&B"N0C%+Á&I&IC5BL("@9C%&)-1C@D$'@."@9C C+9/-12""!#%-0E,3"N0C,U

<,@ #%$'2:/[email protected],c$'2«(*CB5/ /.@.$'BAh'C%@ B"2"[email protected]%N1/>-0$'2 []!%XB"-0h,N0C%2"#C≈C2X/[email protected]%+dÁ$'@9&B"N1C+_/>C%N0N1C XB"C+.-

ϕ ≈ ψc<,N1$,@.+

ϕ ≡ ψcC/ (H$'B"@ B"2

q ∈ NO5A!,c<-1N ,-7/B"2 25$'&3"@9C OE2"-e5C dÁ$'@.&B"N1C+2"$'2!%XB"-7h,N1C2X/.C%+(E,@

≈cZ"C("@9$,dÁ$'2""CB"@D"CXBE,2:/>-7OE#Ü/>-1$,2LB(5N1B"+

qU

lC/9/>Cy!%XB"-0h,N0C%2"#CB5/>-0N1-1+9CKNV[\!aB5-0h,N1C2"#%C_5C%+dÁ$'@.&B5N1C%+3*$a$'N1!C%2"25C%+%cEXB"-øC+9/?"!#%-1",3"N1CU2"C gZ¾ÉqpÂfÈ HÂÁÌXÈAÈLC+9/_B"25CdÁ$'@.&B"N1Cy#%$'25+9/>@9B"-0/.Co (",@9/.-1@?"C%+_#$'2"2"C#/>CB"@.+

∧c∨C=/

¬c<C=/4"C ¿5À"¾ÜfÀ"ÂÁÈAÎLHÂfÌXÈaÈaÎ,U?2"C`ÀEÎÎ%zHÀ5°Á* (H$'B"@B52"C dÁ$'@9&B"N1C)3H$A$'N0!%C2"2"C C+9/4B52"CdÁ$'2"#=/>-0$'2ÆXB"-o`#w",XB"Ch,@.-1,3"N1Cy3H$A$'N0!%C2"2"C4,+9+.-0M'2"Cha@>-<$'BÆdf,BaºU3KC%BaÆdÁ$'@9&B"N1C+K3H$A$,N1!%C2"2"C+

fC/g+.$,2X/Ì<;pr¿"ÀEÂÁÈa*mÈaÎ+.-(H$'B"@K/.$'B5/.C%+N1C+_,+.+9-1M'2"/>-0$'2"+%c

fC%+9/_ha@.,-1C4+9-üC/+9C%B"N0C%&)C%2:/+.-

gC%+/ ha@.,-1CUø[\!aB5-0h,N1C2"#%C)C%2:/[email protected])N1C%+_dÁ$'@9&B"N0C%+_3H$A$'N0!%C2"2"C%+C+9/y"!%#-1E3"N1CUøWYN+.B / (E,@C=AC%&)("N0C"C#%$,&I([email protected]%@DN0C%+e/>,3"N1C%,BA5Ch,!%@9-0/>!"C+"C%BAdÁ$,@.&B"N0C%+UtA$'-7/

MSOL∗0(τ,X,Y)

NV[\C2"+.C&3"N0C4"C%+_dÁ$'@9&B"N1C+_/.$'&I-0XB"C%+5CMSOL0(τ,X,Y)

cºC=/(*$,B"@/>$,B5/q > 0

cMSOL∗

q(τ,X,Y)C%+/ NV[\C2"+.C&3"N0C4"C%+_dÁ$'@9&B"N1C+"C

MSOLq(τ,X,Y)"C4Nb dÁ$,@.&)C

QX,ψcø$

QC%+9/yB"2«XBE,2:/>-7OE#/.C%B"@C=/

X 6∈ X ∪ YB52"C)h,@9-b,35N1C"BÄ("@9C%&)-1C%@_$,B«"B«+.C#%$'2"$'@9"@.CUU$,B5/>CdÁ$'@9&B"N1C

ϕ ∈ MSOLq(τ,X,Y)(*CB5/+%[]!%#@.-1@9C

B(ϕ1, . . . , ϕm)c$

BC+9/B"2"CüdÁ$'@9&B"N0C3H$A$'N0!%C2"2"C_C/?(H$'B"@e/.$'B5/

i ∈ 1, . . . , mcϕi ∈ MSOL∗

q(τ,X,Y)U

tA$'-7/q ∈ N

C=/ϕ ∈ MSOL∗

q(τ,X,Y)U <,@?"!=OE2"-0/.-1$'2L"CyNb ("@.$,dÁ$,2""C%B5@K"C aB",2:/>-0O"#/.-1$'2c

ϕ(*CB5/+%[]!%#%@9-1@9C

QX,B(ψ1, . . . , ψm)cD$

BC%+9/LB"2"C dÁ$'@9&B"N1C^3*$a$'N1!C%2"25CÄC=/(*$'B5@/.$'B5/

i ∈1, . . . , m

cψi ∈ MSOL∗

q−1(τ,X,Y)UÃ2LO5ACB52h,$A#%,3"B"N1,-1@9C

τh'C#

ui

+@a&3H$'N1C+"[email protected]/.-1$'2[\,@9-0/>!icH(H$'B"@D/.$'B5/

i ∈ N∗ U tA$'-0/XB"2 C2"+.C&3"N1C)OE2"-á"C h,@9-b,3"N0C%+"B ([email protected]%&)-1C@ $,@."@9CIC=/

YB"2 C%25+.C%&3"N1C)OE2"-á"CIh,@9-b,3"N0C%+y"B+.C#%$'25ý$'@9"@.CUøC%+4dÁ$'@9&B"N1C+4/.$'&I-0XB"C%+4h,C%#N1C+h,@.-b3"N1C+"B("@.C&I-0C%@$,@."@9C E,2"+

XC=/ N1C+h,@.-1,3"N0C%+_"B«+.C%#$'2" $,@."@9C)+9$'2:/5CIN1dÁ$'@9&IC

x1 = x2cU(x1)

$'BR(x1, . . . xi)

$RC+9/ B"2"[email protected]/.-1$'2uº[Q,@9-0/.!

icU ∈ Y

C%+/B52"C4h,@.-b3"N1C5B«+9C%#%$,2"k$'@9"@.C C=/ (H$'B"@/.$'B5/jcxj ∈ X

U $'2"#|MSOL∗

0(τ,X,Y)| ≤ |X|2 + |Y| × |X|+∑

i>0 ui|X|iU[\!aB5-0h,N1C2"#%C

≈C+9/"!=OE2"-1CL@.!#%B"@9+.-0h,C%&)C%2:/Uta$'-0/

q ∈ NcC=/

ϕ, ψ ∈ MSOL∗0(τ,X,Y)

UtA-q = 0

cD,N0$'@.+ϕ ≈ ψ

+.-ϕ = ψ

UtA-02"$'2cϕC+9/"CuN1kdÁ$'@.&)C

QX,ϕ′ C=/ ψ "C NbkdÁ$'@9&ICQ′X ′, ψ′ U tA- X C%+9/áB"25Ch,@9-b,35N1Ce"B)("@9C%&)-1C%@$,@."@9C,cX,N0$'@.+

ϕ ∈ MSOLq−1(τ,X∪X,Y)c,+9-12"$'2

ϕ ∈ MSOLq−1(τ,X,Y∪X)UMC&)%&)C,c'+9-

X ′ C%+/áB"2"Ch,@.-b3"N1Ce5B ("@9C%&)-1C%@$,@."@9C,cX,N0$'@.+ ψ ∈MSOLq−1(τ,X∪X ′,Y)

cm+.-125$'2ψ ∈ MSOLq−1(τ,X,Y∪X ′)

UN1$'@9+ϕ ≈ ψ

+9-Q = Q′ c X C=/

X ′ +.$'2:/"C%+áh,@.-b3"N1C+"B &)%&)C$,@."@9C,cXC/ ϕ ≈ ψ′ X′

X

U<@G@9!%#%B5@[email protected]"#%Cc5-0NR) B"2I2"$'&3"@9C?O"2"-Z"C

M

#%N1,+.+9C%+[\!XB"-0h,N0C%2"#CG"C≈E,2"+

MSOLq−1(τ,X,Y)UK$'2"#GN1Cå2"$'&[email protected]á"C#Nb,+9+.C%+[]!%XB"-0h,N0C%2"#C"C

≈E,2"+

MSOL∗q(τ,X,Y)

C%+/OE25-GC=/+.C@>[QB ("N1B5+2 × (|MSOLq−1(τ,X ∪ X,Y)| +

|MSOLq−1(τ,X,Y ∪ X)|)U

tA$'-7/ϕ, ψ ∈ MSOLq(τ,X,Y)

UϕC=/ψ(*CB5h'C2X/y+[\!#%@[email protected]@9C%+9(*C#/>-7h'C&IC2X/

B(ϕ1, . . . , ϕm)C=/

B′(ψ1, . . . , ψm′)cmh,C%#

ϕi, ψi ∈ MSOL∗q(τ,X,Y)

U <,@@.!#%B"@[email protected]"#%Cc-0N^ B"2«25$'&3"@9C O"2"-"C#%N1,+.+9C%+[]!%XB"-7h,N1C%25#%C4"C≈E,2"+

MSOL∗q(τ,X,Y)

UøC42"$,&3"@.C"C #%Nb+.+.C+_[]!%XB"-0h,N0C%2"#C4"C≈E2"+

MSOLq(τ,X,Y)C+9/OE2"-øC=/+9C%@.)[\,B("N1B5+

22|MSOL∗q(τ,X,Y)| U

3$ (f&1$ + 4#$ &19$ 2 68D#+M6"1 $tA$'-7/

τB52Æh'$a#,3"B5Nb,[email protected],U*Ã2u"!=OE2"-0/

τ kN1C h,$A#%,3"B"N1,-1@9C #$'2:/>C2E,2:/_/>$'BA/>CN0C%+@9C%N1/>-0$'2"+?"C

τch'C#

[email protected]/.-1$'25+eB"2E,[email protected]+

Pi, . . . Pk

UtA$'-7/AC=/A′ "C%BA τ 6g+9/[email protected]"#=/>B"@9C%+%c,"Ce"$'&I,-02"C%[email protected]+.(HC%#=/>-0dÁ+ A C=/ A′ U'Ã2+9B"("(H$'+.CDXB"C A∩A′ =

∅Á+.-125$'24$'2("@9C%2" B"25Ce#%$'(5-1Ce"-0+Á89$'-02:/>Cá"C

A′ U'[\B52"-1$'2"-1+r89$'-12:/>CG"C A C=/ A′ C%+/Nb τ 6Y+/>@9B"#/.B"@.Cc2"$,/.!%CA⊔A′ ca"C5$'& -12"C A∪A′ caC=//.C%N1N0CaB5CK(H$'B"@á#wE,XB"C@9C%NbÜ/>-1$,2 R c RA

F

A′= RA ∪RA′ U

,25+?N0C%+#+?5C%+M'@>(""C%+cZ#%C/9/>C$'(H!%@./>-0$'2 #%$'@[email protected]+.(H$'2" o4NV[\B"25-1$'2 "-0+Á89$'-02:/>C,U§·i@9C%("@9!%+.C2:/>CNb

τk6Y+9/[email protected]"#=/>B"@9C,c<mh,C%#`#$'&)&IC)"$'&I,-125C

vc/.C%N1N0C aB5C

Pi = vC=//.C%N1N0CXB"C_/>$,B5/>C+?N0C%+?,B5/[email protected]+?@9C%N1/>-0$'2"+e+9$'2:/?ha-0"C%+U§

ρi→j(A)C+9/Nb +9/[email protected]"#=/>B"@9C "Cy"$'&I,-02"C

AcE/>CN1N0C_aB5C

Pρi→j(A)i = ∅

cP

ρi→j(A)j = PA

i ∪ PAj

cC=/XB"C(H$'B"@e/.$'B5/>C+N0C%+,BA/>@[email protected]%N1/>-0$'2"+Rρi→j (A) = RA U§ tA$,-0/

RB52L+ A&3*$'N0C"[email protected]%N1/>-0$'2,B5/[email protected]"C

Pi

ci ∈ 1, . . . , k

c"C=/K+9$'-0/a+.$'2^,@.-7/>!,U`KN1$,@.+

ηR(i1,...,ia)(A)

C+9/KNb +9/>@9B"#/.B"@.C "Cy"$'&I,-12"CAc*h,C%#

RηR(i1,...,ia)

(A)= RA ∪ PA

i1× . . .× PA

iaC=/(H$'B"@e/.$'B5/+ A&3*$'N0C_"[email protected]/.-1$'2QB5/>@9C_aB5C

RcQ

ηR(i1,...,ia)

(A)= QA U§

A⊕A′ C%+/NV[\B52"-1$'2 5-1+Á89$,-12:/>C_"C++9/>@9B"#/.B"@.C+ A C/ A′ cE#,[]C%+9/6oÜ6g"-1@9C A⊔A′ U N1,@.M,C%B"@5C#N1-0aB5C [\B52"Cτ6Y+/>@9B"#/.B"@.C

AC%+9/)N0C ("N1B"+4(HC/.-0/

k/.C%NDXB[]-1NeC+9/)(H$'+.+9-13"N0C`"C#%$,2"+9/[email protected]"[email protected]_Nb

τk6g+9/>@9B"#/.B"@.C

A′ h,C%#yN1C%+$'(*!@>/.-1$'25+D#%-"C+.+9B"+%cZC/D/.C%NXB"C A C%+/Nb+9/[email protected]"#=/>B"@9C A′@.C+9/[email protected]%-02:/>C"C%+?("@9!%"-0#/.+?B52E,-1@9C%+Pi

ci ∈ 1, . . . , k

U

5!%#%$,&I(H$'+.-7/>-0$'2ÆC%2Ä#%$,B"(*C+y,-12"+9-<XB"C Nb`5!%#%$,&I(H$'+.-7/>-0$'2Æ3"-Å6089$'-02X/K+.$'2:/y"CBA M'!2"!%@.,N1-0+>Ü6/>-0$'2"+"CDN1_"!%#$'&I(H$'+9-0/>-0$'2&)$A5B"Nb,[email protected]'!2"!%@.,N1-0+>/.-1$'2E,2"+N0CD+.C%25+GXB[]B"2)&I$a"B"N0Ce[]B"2IM,@>,("5CC%+//.$'BÜ89$'B"@9+ B52"CI#$'B"(HC C=/B"2«3"-76789$'-12:/%U KC #C)df-0/yB"2kM,@>,("5CI#$'&I(5NE /.C%&)C%2:/ "!%#$'&I(H$'+.,3"N1C(E,@GNb"!%#$'&)(*$'+9-0/.-1$'2)&I$a"B"N1,-1@9CD+.C@>#%$,&I("N /.C%&)C%2:/5!%#%$,&I(H$'+>3"N1C?(E,@GNb 5!%#%$,&I(H$'+.-7/>-0$'2 C%2#%$,B"(*C+C=/Nb4"!#%$'&)(*$,+.-0/.-1$'2 3"-76789$'-12:/Ul$'&)&IC,B5/[email protected],!%2"!@>,N0-1+>Ü/>-1$,2I"CNby"!%#$'&I(H$'+9-0/>-0$'2)&I$a"B"N1,-1@9C,c:$'2I(*CB5/°89$'BA/>C%@GNby"!%#$'& 6(H$'+.-7/>-1$,2C2^#$'B"(HC%+K5B^M'@.,(""Cy#%$'&)("N0!%&)C%2:/w,[email protected]"Cy3"-Å6Y(E,@/>-7/>-1$,2 "C%+?+.$'&)&)C/>+[]B"2ÆM,@>,("5CC%+/y,("(HC%N0!%C4B"2"C Í'">Í'mpÇÈÆ+9-ü#%C=/./.C)35-76Y(",@9/.-0/>-0$'2C%+9/B52"C #%$'B"(HC4"B M'@.,(""C4#$'&I(5N1!%&)C%2:/>,-1@9C,UøC%+y3"-76789$'-12:/.+_"Bk#$'&I(5N1!%&)C%2:/>,-1@9C[]B"2 M'@.,(""C +.$,2X/ C=5,#=/>C&IC2X/N1C+y35-76089$,-12:/>+_"B«M'@>(""C,cC=/"$'25#y2º[\-02""B"-0+.C%2:/?(E,+"C2"$,B5h'CN1N1C_5!%#%$,&I(H$'+.-7/>-0$'2U,2"+?#%C#wE("-0/[email protected],cE$,2!=/>B"5-1C"C%+M'!2"!%@.,N1-0+>/.-1$'25+e"C#%C+"!#%$'&)(H$'+.-7/>-1$,2"+%U

jmlkqk tL|k ff_z Kx fg ' I k tL| Gfg o>:

,C,2a6J^,@9#)2"N1-0M'2"C%NVc*",2"+yNb`#%$,2"#%N0B"+.-0$'2 "C +> />! +.CLP\,2ERATVcø(E,@.N0C4-125dÁ$,@.&)C%N1N0C%&)C%2:/"C#%C aB5C`(*$,B"@.@.,-0/y/[email protected]`N1N1,@.M,C%B"@y&I$a"B"N1,-1@9C [\B52 M'@>(""C,cB"2"C +.$'@/>C`"CIM'!%25!%@>N1-1+./>-0$'2Ä"C N1"!#%$'&)(*$,+.-0/.-1$'2 &)$A5B"Nb,[email protected],WYNE"-7/GXB"C?#%C(E,@.,& =/>@9C?(H$'B"@9@>,-7/å/[email protected]?B"2"C,N0/.C%@.2"/>-7h'Co N1_Nb,@9M'C%B"@"C#%N0-1XB"C_C=/o4N1Nb,@9M'C%B"@ ;WlKU,2"+ #C/9/>C +.C#/>-0$'2cG$'2 "$'2"25CuB"2"Cu"!=OE2"-0/.-1$'2ýdÁ$'@.&)C%N0N1C[\B52"CuNb,@9M'C%B5@`&I$a"B"N1,-1@9C,cå,-02"+.-XB"C`"[email protected]/.-1$'25+ "C #%C/9/>C`N1,@.M,C%B"@ h'C%# NbLN1,@.M,C%B"@ "C #N1-1XB"CIC/NbLN1,@.M,C%B"@;KWlKUÃ2kh,C%@9@>XB[]-1N+[Q,M,-0/D"C(E@>,&g /[email protected]%+D+9-1&)-1N1,-1@9C%+%cXC=/XB"CN0C_@>,("(H$'@/DC%2:/>@9C #C%[email protected]'CB"@.+DC+9/,B("N1B5+

2UZÃ2"$'252"C%@.4,B"+.+9-B"2"C_25$'B5h'CN1N0C,N0M %3"@9C(H$'B"@#$'2"+/>@[email protected]_B52 M'@>,(5"C"CN1,@.M,C%B"@G&I$a"B"N1,-1@9C

kmh,C%#,B("N0B"+

k#%$'B"N0C%B"@9+"-*!%@9C%2:/>C+%U

#$g2k . ,/&$g2

Ì ÁmfH ;()+()(-tA$,-0/k ∈ N∗ c G = (V,E)

B"2ÄM'@>,(5"C4C/V ′ ⊆ V

UÃ2 "-0/_XB"CV ′ C%+/yB"2

k>ÉkøËWpÂfÈL+%[]-1N<C A-1+9/.C B"2"C (E,@/>-0/.-1$'2

V ′1 , . . . , V

′k"C

V ′ />CN1N0CXB"C (*$,B"@/>$'BA/ i ∈ 1, . . . , k cC/?(H$'B"@e/.$'B5/u, v ∈ V ′

i

cN(u) \ V ′ = N(v) \ V ′ U

u

" ¢¡Z£ "£7¥¦ § 236g&I$a"B"N0CKE2"+B"2LM,@>,("5C h,C%#y+> (E,@9/.-0/.-1$'2U

Ã2 h'$'-7/df,#%-0N1C&IC2X/XB[\B5216Y&)$A5B"N1C)#%$'@[email protected]+.(H$'2"so^B"2&)$A"B5N1C,U^OEM'B"@9CGx5U7 "$'2"25C`C%2C=AC&I("N0CB"2

36Y&)$a"B"N1CKE,2"+B"2M'@>(""C,UÃ2 h,C%@9@>E,2"+Nb +9C%#=/>-1$,2 xAUQS #%$,&I&)C%2:/ $'2 (HC%B5/5!%#%-0"C%@C2 /.C%&)("+(*$,N<a2"$'&)-b,N+9-_B"2M'@.,(""C(H$'+9+7 %5C_B"2

k6g&I$a"B"N0C?25$'2`/>@9-0ha-b,NVc:(*$,B"@

kO5A!UEÃ2`h'C@.@>!M:,N0C%&)C%2:/eXB[]B"2"C_+/>@.B5#/>[email protected]@.C("@.!+.C2X/>,2:/D/>$'B5+N1C+

k6Y&)$A"B5N1C%+á"B`M,@>,("5C(*CB5/e=/>@9CK#N1#%B5N1!%CC%2 /.C%&)("+e(H$'N<a2"$,&I-1,Nôc(*$'B5@

kO5A!,U2"CIÍ:ÂÁÀEÎÎ%ÈÆ",2"+G[]B"2k+9$'B"+6gC%2"+9C%&35N1C

V ′ 5C V C%+/B"2Ä+.$,B"+6gC%2"+9C%&3"N0CI&IÜA-1&I,NC"C+.$,&I&)C/.+"C

V ′ /.C%NºXB"CK(*$'B5@á/.$'B5/ u ∈ V (G) \V ′ ca+.$,-0/ u C+9/Ü8.,#%C2:/oy/.$'B"+eN0C%++9$'&)&IC=/>+á"CCc:+.$'-7/yoy,B"#B"2UAÃ2I25$,/>C@>(E,@

PartG(V ′)Nô[]C%25+.C%&3"N1C"C+G#Nb,+9+.C%+á"C

V ′ E,2"+ G UaÃ2)h,$'-0/áXB"CV ′ C%+/4B"2 k 6Y&)$A5B"N1C +.-áC/+.CB"N1C&IC2X/+.- |PartG(V ′)| ≤ k

UøC%+ "C%BAk$'3"+9C%@h,Ü/>-1$,2"+ +9B"-0h,2:/.C%++.$,2X/?-0&I&)!%"-1/>C+%UåÈAÉkÉkÈ ;()+(@

V ′ DNL]IKH1 k R>/QF?KHBED FHD G y:?BE/[LV ′ DNL]I KH1 k R>/QF?KHBED FHD G c

åÈAÉkÉkÈ ;(E;(@V ′′ ⊆ V ′ ⊆ V

D]IV ′′ DNL]IyKH1 k R>/QF?KHBEDFHD G :?BE/[L

V ′′ DNL]IyKH1 k R>/QF?KXBED$FHDG[V ′]

ctA-V1, V2

C+9/<B"2"Ce#$'B"(HC,c,B52"C#$6Y#$'B"(HCe$'BB"23"-76789$'-12:/c°N1$'@9+V1

C/V2

+9$'2:/"C%+26Y&)$a"B"N1C+%U

K$'25# +9-V ′ C+9/KB"2L&)$a"B"N1Cc V \ V ′ C%+/KB52 2

6g&I$a"B"N0C,U åN0B"+M'!%25!%@>N1C%&)C%2:/$'2(*CB5/K5!%&)$'2:/>@.C@N1C_N0C%&)&ICK+.B"-7h,2X/%UåÈAÉkÉkÈ ;( ;(@

V ′ DNL]IKH1 k R>/QF?KHBED:?BE/[L V \ V ′ DNL]I KX1 2kR>/QF?KHBEDQc

]>/1`L]I[:?I*@/1mctA$'-7/u 6∈ V ′ UKN0$'@.+(H$'B"@#wE,XB"C #%Nb+.+.C C "C

V ′ c u C+9/+.$'-7/,Ü8.,#C%2:/ o />$'B5+N1C++.$'&)&IC=/>+D"CCcE+.$'-7/oI,B5#%B"2U l$'&)&)C_-1NK),B("N1B"+

k#%N1,+.+9C%+?E,2"+

V ′ cE-1NK),B("N1B"+ 2k,Ü8.#%C%25#%C%+?(H$'+.+9-13"N0C%+e(H$'B"@DN0C%+D+.$,&I&)C/.+D"CV \ V ′ E,2"+ V ′ UK$'25# |PartG(V \ V ′)| ≤ 2k

UFZ,3"-1C2k"C`JL$'2:/>M,$'N0OEC@#%$'2"+.,#%@9!`B"2 #w",("-0/[email protected]"C +.L/>! +.CGoLNô[]!/>B5"C "C+

26g&I$a"B"N0C%+UøWYN&)$'2:/>@.C_C2L(",@9/.-1#%B5N1-1C@eXB[]-1N1+D25CdÁ$,@.&)C%2:/(E,+cEC%2M'!%2"!@>,NVc"B"2"C_df,&)-1N0N1C?(E,@/>-0/.-0h,C,U

lC%N1C+9/y"!Y87oB52 &I,B5h,-0+(*$,-12:/ Æ(*$,B"@_Nb`"!=OE2"-7/>-1$,2u"C4#%C XB"C (*$,B"@.@.,-0/K/>@9C4B"2"C "! 6#%$,&I(H$'+.-7/>-0$'2k6Y&)$a"B"[email protected]

Nb)("@.$'([email protected]!=/>!y"C(",@9/.-0/>-7ha-0/>!(HC%@9&IC=/./w-0/?[\mh,$'-1@?Nô[]B"2"-0#%-7/>!"CN1I"! 6#%$,&I(H$'+.-7/>-0$'2UN1N0CM:@>,2:/>-7/eB"+.+9-ºXB[]B"2"C"!#%$'&)(H$'+.-7/>-1$,2 @.!#%B"@9+.-7h'C(E@B"2 &)$A5B"N1C2"$'2`/[email protected],N,3H$'B5/.-0/e/.$'BÜ89$'B"@9+o N1"!%#$'&)(*$'+9-0/.-1$'2U

k ! 7 / !

Ã2(*CB5/ü25!,2"&)$'-02"++.B"M'M,!%@.C@oKXB"$,-a#$'@.@9C%+9(*$'25"@>,-7/<B52"Ce"!#%$'&)(*$,+.-0/.-1$'2k6Y&)$A"B5Nb,[email protected]ü[]B"2M'@.,(""CU

Ì ÁmfH ;()+( (tA$'-7/GB"2 M'@.,(""CU2"C`ËÌaÍ'HÉkÇHÎr°fH

kwÉ«øË pÂÁÀEf¾°È "C

GC%+/4B"2"Cdf,&)-1N0N1C

F[]C%2"+9C%&3"N0C%+?"C+.$'&)&)C/>+D5BLM'@.,(""C/.C%N0N1C_XB"C9§ (H$'B"@e/.$'B5/+.$,&I&)C/

v"BLM,@>,("5C,c

v ∈ Fc§

V ∈ Fc§ (H$'B"@e/.$'B5/

X ∈ FC=/Y ∈ F

cX2"C#w"C=h,B"#w"C (E,+

Yc§ (H$'B"@e/.$'B5/

X ∈ FcX-12""B5-0/B"2

k6g&I$a"B"N0C"C

GU

,2"+N1Cy#,+?"C+16g&I$a"B"N0C%+%c5#%C/9/>Cydf,&I-0N1N0CK#%$,@[email protected]+.(H$'2"uBAL&I$a"B"N0C%+DdÁ$'@9/.+5B^M'@.,(""CU,@"!=OE2"-0/.-1$'2ºcºB"2"C45!%#%$,&I(H$'+.-7/>-0$'2

k6g&I$a"B"N1,-1@9C º[\B"2 M'@.,(""C4C+9/yB"2"Cdf&I-0N1N1C ,@93*$'@9C%+9#%C%2:/.C,U N1N0C"C& ,25"CC24("N1B"+XB"Ce#wEaB5CD&IC&3"@.Cá"CeN1df&I-0N1N1C-02""B"-0+.CáB"2

k6Y&)$A"B5N1C,U°?2"Ce"!#%$'&)(H$'+.-7/>-1$,2

k6g&I$a"B"N1,-1@9C[\B52M,@>,("5C#%$'@[email protected]+.(H$'2" "$'25# oB"2,@[email protected],ca/>CNXB[\-0N_`,-0/B"2"C35-¢89C%#=/>-0$'2`C%2:/[email protected]_N1C++.$,&I&)C/.+D"BLM,@>,("5CyC=/?N1C%+edÁCB"-1N0N1C+e"CyNô[\,@.3"@9C,c"C=/?(*$'B5@e/>$'B5/25$ACB"L5C NV[Q,@93"@.Cc"Nô[]C%25+.C%&3"N1C_"C+dÁC%B5-1N1N0C%+_"B«+.$'B5+6ö@.3"@9CIC2"@>,#-12"!),B«2"$aC%B"«-12""B5-0/B"2

k6Y&)$a"B"N1C"C

GUÃ2 ,("(HC%N0N1C4#C/,@.3"@9CÂ7À5¾Ü¾°ÈÆËÈuÂÁÀ«ËÌaÍ'HÉkÇHÎÁmfHåUÃ2hÆ+[\-02X/.!%@9C%+.+9C%@oÆ(E,@/>-1@"C & -12:/>C2E,2:/ ,BA M'@>(""C%+4#$'&)("NE =/>C%&)C%2:/45!%#%$,&I(H$'+>3"N1C+(E,@D#C/9/>Cy"!%#$'&I(H$'+9-0/>-0$'2

k6Y&)$A"B5Nb,[email protected],U

*^+ ,.)01,2547684,9 0 #k . ,/&$g2 $g6 #$ (h. ,/&(4B%$

Ì ÁmfH ;(*)+( ;(tA$'-7/G = (V,E)

B"2uM'@.,(""CUøÃ2Æ"-7/XB"CGC+9/yËÌAÍ'HÉkÇHÎ%À5<ÂÁÈ ÇÀ"¾

kÉkøËWpÂfÈaÎå+%[]-1NXC=A-1+/>CGB"2,@93"@.Cá3"-02E,-1@9CC%2"@.,#%-02"!

T/>C%NXXB"CG-1N ,-0/B"25CG3"- 89C%#=/>-1$,2 C2X/[email protected]%+dÁC%B5-1N1N0C%+"CNô[\,@.3"@9CC/eN0C%+á+.$'&)&)C/>+å"B M'@>,(5"C,cXC/GXB"C(H$'B"@å#wE,XB"CK2"$ACB"

a5CKNô[\,@93"@.Cc:Nô[]C%2"+9C%&3"N0C5C%++.$,&I&)C/.+D#%$'@[email protected]%+9(*$,2"E,2:/,BA dÁCB"-1N0N1C+e"B+.$'B"+ö6ö,@93"@.CC2"@>,#-12"!C2

aC%+9/?B"2

k6g&I$a"B"N0C,U

Ã2^h'$'-7/3"-0C%2LaBº[\-0N+[Q,M,-0/[\B52u#,+(E,@9/.-1#B"N1-0C%@"CN1I"!#%$'&)(H$'+.-7/>-1$,2k6Y&)$A"B5Nb,[email protected],cA/>C%NXB"C#wE,XB"C2"$aC%B"Ï"CNô[\,@93"@.C "C"!#%$'&)(H$'+.-7/>-1$,2 ,Bý(5N1B"+4"CBAOEN0+%U (E@9/>-0@4"C& -12:/>C2E,2:/cXBE,2" $,2 (E,@.N0C%@. "C5!%#%$,&I(H$'+.-7/>-0$'2

k6Y&)$A"B5Nb,[email protected],c,$'2 +.$'B5+eC%2:/>C2""@. XB"CK#wE,XB"C2"$aC%B"`",2"+Nô[\,@93"@.C5C4"!%#$'&I(H$'+9-0/>-0$'2u ,B ("N0B"+

2OEN0+%UºC+_M'@>(""C%+"!#%$'&)(H$'+>,35N1C%+K(E,@

16g&I$a"B"N0C%++.$'2:/C=5,#=/>C&IC2X/ N1C+I#$'M'@>(""C%+UeÃ2ý@.C/.$'&3HC"$'2"#L3"-1C2ý+9B"@ N1Ä#Nb,+9+.C"C%+IM'@>,(5"C%+IC%2:/>- %@.C&IC2:/"!#%$'&)(*$,+>,3"N0C%+D(E@?N1"!%#$'&)(*$'+9-0/.-1$'2 &)$A5B"Nb,[email protected]Ã2L(HC%B5/?"!=OE2"-0@DB"2L(",@>,&g /[email protected]_"CM'@.,(""C#$'@.@9C%+.(H$'2"",2:/o4#C/./.C 5!%#%$,&I(H$'+.-7/>-0$'2

Ì ÁmfH ;()( &(tA$'-7/G = (V,E)

B"2«M'@>,(5"C,UÂfÀ"¾zZÈHp¾ É«øË pÂÁÀEf¾°ÈL"CGC%+/N0C)(5N1B"+(HC/>-7/

k ∈ N∗ />CNøXB"C G +.$'-7/"!%#$'&)(*$'+.,3"N0C_(E,@ k 6g&I$a"B"N0C%+%U5WYNC%+/K2"$/>! mod-wd(G)U

øC%+M'@>,(5"C%+"-0+9/w2"#%Cu"!@.!"-0/>,-1@9C,ceN0C%B"@ #$'&I(5N1!%&)C%2:/C=/N1C%+M'@.,(""C++>,2"+C5

ce/>,B"@9C,BcM'C&C=/ #%$6gM'C%&-125"B"-0/$'2:/4B"2"C N1,@.M,C%B"@ &I$a"B"N1,-1@9C)º[Q,B (5N1B"+2U2 C *C/%cü#%C+4M'@>(""C%++.$'2:/#%$,&I("N /.C%&)C%2:/4"!#%$'&)(*$,+>,3"N0C%+(",@4Nb^5!%#%$,&I(H$'+.-7/>-0$'2 C%2#$'B"(HC%+%c<NbÆ"!#%$'&)(*$,+.-0/.-1$'2«3"-76789$'-12:/$'BuN1`"!%#$'&I(H$'+9-0/>-0$'2^C2 #%$6g#%$'B"(HC%+UWtA-<NV[\$,2uC%2"@.,#%-02"C4Nô[\,@[email protected]"!%#$'&I(H$'+9-0/>-0$'2cHC/$'2u5-0ha-1+9C/>$,B"+_N1C+_2"$ACB""+-12:/.C%@.25C%+C%2Ä2"$aC%B""+_5C)5C%M'@9!

3cºNV[Q,@93"@.C4$,35/>C2aBÄ-125"B"-1@. B"2"C "!#%$'&)(H$'+.-7/>-1$,2

26g&I$a"B"N1,-1@9C,U,J^,-0+Nb_#%N1,+.+9C?5C%+áM'@>,(5"C%+á"CNb,@9M'C%B5@&I$a"B"N1,-1@9Ce,B)("N1B5+

2C%+/á+9/>@9-1#=/>C%&)C%2:/á("N0B"+M'@.,2""CXB"CNô[]B"2"-0$'2 "C#%C+?/.@.$'-0+D#%Nb+.+.C+"CM'@>,(5"C%+cE#,@C%N0N1C#$'2:/>-0C%2:/cE(E@?C AC%&)("N1Cc"N1C

C5U

u

b

dc

e

a

b c

a

ed5 ]¡Z£ "£T|¦§ ?2"C"!%#$'&)(*$'+9-0/.-1$'2

26Y&)$A"B5Nb,[email protected]?"B

C5

UåÈAÉkÉkÈ ;( &(;/@CI

GKX1PY?[:]\+^+DQc;BE/[L

mod-wd(G) = mod-wd(G)c

]>/1`L]I[:?I*@/1mcDWY&I&)!%5-b/%c'#,@cX(E,@áN1C?N1C%&)&)Cx5U0,c:B"2"CK5!%#%$,&I(H$'+.-7/>-0$'2k6Y&)$a"B"[email protected]"C

GC+9/B"2"C"!#%$'&)(H$'+.-7/>-1$,2

k6Y&)$a"B"[email protected]?"C

GU

åÈAÉkÉkÈ ;(<;;(;/@CIG = (V,E)

KX1PY?[:]\+^+DV ′ ⊆ V

c BE/[Lmod-wd(G[V ′]) ≤ mod-wd(G)

c]>/1`L]I[:?I*@/1mctA$'-7/

FB"2"C)"!%#$'&)(*$'+9-0/.-1$'2

k6Y&)$A"B5Nb,[email protected]

G/>C%N0N1CXB"C)Nô[\,@93"@.C4"C)"!#%$'&46(H$'+.-7/>-1$,2Ä+9$'-0/ 3"-02E,-1@9C,Udf&I-0N1N1C

F ′ = X ∩ V ′ : X ∈ FC/X ∩ V ′ 6= ∅

C+9/B52"CIdf,&)-1N0N1C,@93*$'@9C%+9#%C%2:/.C,cø"$'2:/NV[Q,@93"@.C[\-02"#%N0B"+.-0$'2u"C4N1Idf,&)-1N1N0C C+9/3"-02E,[email protected],U KC ("N0B"+_(E,@N0CN1C&I&)CxAUTnAc#wE,XB"Cy&)C%&3"@9C_"CF ′ C+9/?B"2 k 6Y&)$a"B"N1C"C G[V ′]

UlC (",@>,&g /[email protected]_C+9/?C%2df,-0/e+.-1&)-1N1,-1@9CKBA(E,@.,&g />@9C%+D"!Y87o)@9C%2"#$'2:/>@9!%+%U

"$# ÌXZ¾%aÉkÈ ;( ;( ;/@CIG

KH1 Y?[:]\+^+DQc BE/[NLmod-wd(G) ≤ NLC-wd(G) ≤ cwd(G) ≤ 2 ×

mod-wd(G)c

]>/1`L]I[:?I*@/1mcDWYNøC%+/5!ö87o #%$'2"2XBXB"CNLC-wd(G) ≤ cwd(G)

P ,$'°TôUtA$'-7/

kN1N1,@.M,C%B"@ ; l"C

GC/tB"2"C

k6YC=A("@9C%+9+.-1$,2 ;KWlý5C

GUHtA$'-7/

t′B"2"C+.$,B"+6gC=A("@.C+.+9-1$'2"C

tcEC/

V ′ NV[\C2"+.C&3"N0C_"C%++.$,&I&)C/.+,("(E,@.,-1+9+>,2:/E2"+ t′ UZÃ2h,$'-0/Ddf#%-1N0C%&)C%2:/DXB"C V ′ C+9/B"2k6g&I$a"B"N0CK"C

GU82LCºC=/c"/.$'B"+DN1C+D+.$'&)&)C/>+D5Cy&)%&)C!=/>-1XB"C=/./.Cy",2"+DN1C_M,@>,("5C_C=A("@.-0&I!_(",@

t′)N1Cy&I&ICh'$,-1+.-02E,M'CyE,25+

V \ V ′ U3K$,2"# T -02""B"-7/B52u,@[email protected],[email protected]_/>CNXB"C Nô[]C%2"+9C%&35N1Cy"C/>$,B5/>C+?N0C%+DdÁC%B5-1N1N0C%+G[\B52^+.$,B"+6Y,@.3"@9CC%+9/?B"2&)$A"B5N1Cmod-wd(G) ≤ NLC-wd(G)

UtA$'-7/

TB"2Ä,@.3"@9C3"-02E,-1@9C /.C%NüXB"C4(H$'B"@K/.$'B5/_+9$'B"+ö6ö,@93"@.CcNô[]C%2"+9C%&35N1C"C+dÁC%B"-0N1N0C%+KC+9/B"2

k6&)$A"B5N1C,U;tA$'-7/

aB"2Ä2"$aC%B"Ä"C

Tc$'2Ä2"$,/.C

Va

NV[\C2"+.C&3"N1C "C+_dÁC%B"-0N1N1C+"BÄ+.$'B5+6ö@.3"@9C)C2"@>#%-125!C%2aUeÃ2 #%$'2"+/>@9B"-0/ @9!%#B"@.+9-0h'C&IC2:/B"2"C

2k6YC 5([email protected]%+9+.-0$'2

ta"C

G[Va](H$'B"@I/.$'B5/ 2"$ACB"

a"CNô[\,@93"@.CcA/>CN1N0CXB"C(*$'B5@e/>$'B5/

u ∈ Va

clabG(ta)(u) ∈ 1, . . . , k

C/(H$'B"@/.$'B5/u, v ∈ Va

/>CN1N0CXB"CN(u) \ Va 6= N(v) \ Va

clabG(ta)(u) 6= labG(ta)(v)

U*Ã2u+.B"("(H$'+9C+>,25+(*C@9/.C"CM,!%2"!@>,N0-0/>!yXB"CN1C+D2"$aC%B""+?-12:/.C%@.25C%+$'2://>$'BÜ89$,B"@.+D"CBAOEN0+%UtA-

aC%+/ÆB"25CÄdÁCB"-1N0N1C,ce,N1$'@9+Nô[]C=A("@.C+.+9-1$'2½C%+9/L/>@9-0ha-b,N0C,U tA-12"$,2c+.$'-7/

bC=/

c+9C%+LO"N1+%UN1$'@9+

Va = Vb ∪ Vc

UtA$,-0/tbC/tcNô[]C=A("@9C%+9+.-1$,2 (*$'B5@

G[Vb]C=/G[Vc]

[email protected]+.(HC%#/.-0h,C%&)C%2:/Uta$'-0/t′′a =

tb + i∈1,...,kρi→i+k(tc))Á$@9C%("@9!%+9C%2:/>CuN1«#%$,&I(H$'+.-7/>-0$'2 =Ul$'&I&)C

Vb

C=/Vc

+.$'2:/ 5C%+k6&)$A"B5N1C%+cø$'2k(HC%BA/#$'2"+9/[email protected]"[email protected])B"2"CIC=A("@.C+.+9-1$'2

t′a"C("B"-0+

t′′aC%2k°89$'B5/>,2:/"C+,@9/.C%+C2:/>@.CIN1C+!/.-1XB"C=/./>C+"C

1, . . . kC=/"C+ !=/>-0aB5C/./.C%+ "C

k + 1, . . . , 2kUWl$'&I&)C $'2Ä2[Qm89$'B5/>C)XB"C 5C%+

k ! 7 / !

,@9/>C+C2X/[email protected]`5C%+!=/>-0aB5C/./.C%+ "C1, . . . k

C=/"C+!/.-1XB"C=/./>C+"Ck + 1, . . . , 2k

c$'2 2[\°89$'B5/.C(E,+ [Q@./.C%+`C2:/>@.C^"C%BA +.$,&I&)C/.+ "CVb

$'B C%2:/>@9CÆ"C%Ba +.$,&I&)C/.+I"CVc

Ut′aC+9/ 5$'2"#^B52"CC=A("@9C%+9+.-1$,2 (*$,B"@

G[Va]U l$'&I&)C

Va

C%+9/ B"2k6Y&)$a"B"N1Ccº-0N LBk("N0B"+

kh,$'-1+9-12E,M,C%+(*$,+.+.-03"N1C+(H$'B"@ N1C+)+9$'&)&IC=/>+ "C

V \ Va

E,25+Va

UÃ2Ï(*CB5/ ,+.+9$A#-1C%@ N0C%+ !/.-1XB"C/9/>C+I"C1, . . . , 2k

C2k#%N1,+.+9C%+%cå#%,@4+.-

u, v ∈ Vb

2"C+.$'2:/)(E,+ E2"+IN1Æ&I&IC #Nb,+9+.C"CVa

cå,N0$'@.+uC/v2"C+.$'2:/(E,+KE,2"+N1 &)%&)C #Nb,+9+.C"C

Vb

Á"C&)%&)C(H$'B"@Vc

=UÃ2Æ#%$'25+9/>@9B"-0/Nô[]C=A("@.C+.+9-1$'2ta"C%(5B"-1+

t′a/>CN1N0C4aB5CI(H$'B"@/.$'B5/u ∈ Va

clabG(ta)(u) ∈ 1, . . . , k

cøC/yN1C+y5-*!%@.C2:/>C%+y#%N1,+.+9C%+y[]!/>-0XB"C/9/>C%+#%$,@[email protected]+.(H$'2""C2:/,Ba"-*!%@9C%2:/>C+#%Nb+.+.C+5Bk6Y&)$A"B5N1C

Va

U`ÈaÉkÀ"¾B;pÈ( 1 13D\+D]KHI\`:L\m[/KHb?D][FHDNLqYM:?B @CINLqFM:?1`L BED IC^+N/[]>D9\m[NbaN7FHD]1+I

BEDP10

:KH13DAB :?[*YHD]KX[$>/QF?KXB :?@C[DAFHD

2D]IKH13DAB :?[*YHD]KX[AFHDJa]B @BAQK+D9FHD

3@a7:?[ @CBWL+V :Y?@CIKH1Y?[:]\+^+D9F?@EL]I:?1KaND

^+][7F?@CI:?@C[DD]I(aNDi13V DNL]I;\`:LiKX1 aN/7Y?[:]\+^+D D]I LQ:B :?[*YHD]KH[-8 DNL]IyFHD3c

0#%$g2 2_4,/9 01, 1 <$^2 (0$^2 &1*^+-,/.)01,2?(! <$^2 0 k . ,/&<$^2

[Q,@93"@.C-12"5B"-1+.,2:/eN1C+k6Y&)$A5B"N1C+G[]B"2 M'@.,(""C"CKN1,@.M,C%B"@á&I$a"B"N1,-1@9C

k2"C25$'B"+5$'2"2"C(",+"[email protected]%#=/>C&IC2X/åB52I&)$H'C%2 "C@9C%#%$,2"+9/[email protected]"[email protected],@>,("5C,U'WYN"+9C%@.,-0/å-02X/.!%@9C%+.+.,2:/á[Qh,$'-1@B"25C?C A("@.C+.+.-0$'2"!=OE2"-1+9+>,2:/N0CM'@.,(""Cc#$'&)&ICá(*$,B"@<[email protected]'CB"@5Ce#%N0-1XB"CC=/ü[email protected]'CB"@;WlKU $,B"@ü#%CNb5cÜ$'2(HC%BA/-12:/.@.$a"B"-1@9C B"2"C 25$'B5h'CN1N0CI$'(H!%@./>-0$'2 +.B"@ N1C+ M'@.,(""C+!=/>-1XB"C=/>!+%UlC=/./.C $'(H!%@./>-0$'2Ädf-0/Nô[]B"2"-0$'2c°89$'BA/>C_N1C+,@.=/>C+?C2:/>@.CN0C%+n#%$,&I(H$'+>2X/.C%+cEC/@.C 6Y!=/>-1XB"C=/./.C_N1C%+n#$'&I(H$'+.,2:/>C%+c"C%2L,("(5N1-1XBE2X/"CBAdÁ$'2"#=/>-1$,2"+D"-*!%@9C%2:/>C+(*$'B5@D#wE,#%B52"C"C+#%$'&)(*$,+>,2:/>C+%U

Ì ÁmfH ;()+(<; ( tA$,-0/k ∈ N∗ U tA$,-0/ G1 = (V1, E1, lab1)

C/G2 = (V2, E2, lab2)

"C%BAM'@.,(""C+L!=/>-1XB"C=/>!+%cS ⊆ 1, . . . , k2

cDC/L(*$'B5@i ∈ 1, 2

cRi : 1, . . . , k → 1, . . . , k

UKN0$'@.+

G1 ×S,R1,R2 G2 = (V,E, lab)h,C%#

V = V1 ∪ V2

cE = E1 ∪ E2 ∪

u, v : u ∈ V1, v ∈ V2, (lab1(u), lab2(v)) ∈ S

C/lab(u) =

R1(lab1(u))

+.-u ∈ V1,

R2(lab2(u))+.-u ∈ V2.

"$# ÌXZ¾%aÉkÈ ;( (;/@CIk ∈ N∗ c ;/@CI MODk

B : a]B :LNLD FHDY?[:]\+^+DNL FH 1+@D_[ba]KH[NL]@Cb?D]>D]1+I \+/KH[iI/KHI

i ∈ 1, . . . , k·i ∈MODk

L/@CIG ∈MODk

D]IH ∈MODk

c BE/[L\+/KH[ I/KHIS ⊆ 1, . . . , k2

R1 : 1, . . . , k →

1, . . . , kD]IR2 : 1, . . . , k → 1, . . . , k

y:?BE/[LG×S,R1,R2 H ∈MODk

;/@CIG

KX1PY?[:]\+^+DQc3BE/[Lmod-wd(G) ≤ k

L]@ D]ILD]KHBED]>D]1+IL]@ @CB D@? @EL]ID KH1 Y?[:]\+^+D]I@BAQKD]IG′ ∈MODk

ID]B AQK+DG

DNL]I@EL/>/[\+^+D G′ LQ:?1`L LDNLA]I@BAQK+D]I*IDNLQc

]>/1`L]I[:?I*@/1mctA$'-7/G′ ∈ MODk

cüC/ +9-tC+9/ B"25C

k6YC 5([email protected]%+9+.-0$'2ý(H$'B"@N1C M'@.,(""C

G′ c,N0$'@.+(H$'B"@ #wE,XB"CL+.$'B"+ö6YC 5([email protected]%+9+.-0$'2t′"C

tcáNV[\C2"+.C&3"N0C5C%+)+.$'&)&IC=/>+ ("@9!%+.C2:/>+ ",2"+

t′-02""B"-7/ B"2

k6g&I$a"B"N0C,U`K$'2"#yNbdf,&)-1N0N1C?"C%+?C%2"+9C%&35N1C%+?"C+?25$ACB""+?("@.!+.C%2:/.+KE,25+N0C%+D+.$,B"+6gC=A("@.C+.+9-1$'2"+"Ct-02""B"-7/DB"2"C"!#%$'&)(H$'+.-7/>-1$,2

k6Y&)$a"B"[email protected]?"C

G′ cEC/ mod-wd(G) ≤ kU

u

tA$'-7/G"[email protected]'CB"@ü&)$A"B5Nb,[email protected]

kc:C/FB52"C"!%#$'&I(H$'+9-0/>-0$'2

k6g&I$a"B"N1,-1@9CG"C

GUaÃ2 #%$'2"+/>@9B"-0/@.!#%B"@9+.-7h'C%&)C%2:/%cº(H$'B"@/>$'B5/

Va ∈ FcZB"2"C

k6YC A("@.C+.+.-0$'2

ta(*$'B5@

G[Va]/.C%N1N0CaB5C+.-

u, v ∈ Va

2"C+.$,2X/?(E+DE,2"+Nb&)%&)C_#%N1,+.+9C_"CVa

cE,N0$'@.+Nô[]!/>-0XB"C/9/>C_"CuC%+9/?"-+º[email protected]/.C"CNô[]!/>-0XB"C/9/>C_"C

vE,25+G(ta)

Umta-|Va| = 1

cta = ·1

UmtA-02"$'2+.$'-7/Vb, Vc ∈ F

/>CN1+XB"CVb ∩ Vc = ∅

C/Vb ∪ Vc = Va

cC/+.$'-7/tbC/tcN0C%+C 5([email protected]%+9+.-0$'2"+(*$,B"@

G[Vb]C=/G[Vc]

U;tA$'-7/S ⊆ 1, . . . , k2

/>C%N<aB5C(i, j) ∈ S+.-C=/+9C%B"N0C%&)C%2:/+.-øN1C++.$'&)&IC=/>+!/.-1XB"C=/>!%+

iE,2"+

Vb

+.$,2X/,°8.,#%C2X/.+,Ba+.$'&)&)C/>+!=/>-1XB"C=/>!+jE,25+

Vc

U+l$'&)&ICVa

C+9/DB"2k6Y&)$a"B"N1CcXC/eXB"CK+.-

u, v ∈ Va

+.$,2X/eE,2"+G"C%BA`#%N1,+.+9C%+e"-+º[email protected]/.C%+c,N0$'@.+4-0N1+4$'2:/I"C%+)!/.-1XB"C/9/>C+ "-+º[email protected]%2:/.C%+%cå$'2Ï(HC%B5/)/>@9$'B5h'C@Rb : 1, . . . , k → 1, . . . , k

C=/Rc : 1, . . . , k → 1, . . . , k

/>CN1+`XB"C^(*$'B5@ />$'BA/u ∈ Vb

!/.-1XB"C=/>!iC=/v ∈ Vc

!/.-1XB"C/.!jc$'2ý

Rb(i) = Rc(j)+9-C/)+.CB"N1C&IC2X/I+.-

uC=/v+.$'2:/IE,2"+)Nbu&)%&)C #%N1,+.+9CL"C

Va

U KN0$'@.+ta = tb ×S,Rb,Rc

tcC+9/?3"-0C%2Nô[]C=A("@9C%+.+9-1$'2"C

G[Va]mh,C%#yN1C+?(5@.$'("@9-1!=/>!%+D5C%&I,2""!C%+%U

l$'&)&IC C=AC&I("N0C,cNô[]C=A("@9C%+9+.-1$,2k+.B5-0h,2:/>C C%+/B"25C26YC=A("@9C%+9+.-1$,2k(H$'B"@N1C

C5

UøC%+dÁ$'2"#/.-1$'25++.$,2X/[email protected]("@.!+.C2X/.!%C+e(E,@åN1CB"@C%25+.C%&3"N1C?"C#%$'B5("N1C+áh,N0C%B"@9+ § -0& ,M,C%+%cC/RI

@9C%("@9!%+9C%2:/>CNbdÁ$'2"#=/>-1$,2-15C%2:/>-7/>!,U

t = ·1 ×(1,1),RI ,RI

((·1 ×(1,2),RI ,RI

·2)×(1,2),(1,2),(2,1),RI(·1 ×(1,2),RI ,RI

·2))

$ 684<,/9 $+ &(*+-,/.)01,32547684,/9 . ,/&(4B%$ $ 6 ! 4 e,4 9 6)N1,@.M'CB"@e&)$A5B"Nb,[email protected] N0C_&I&IC_#$'&)(*$'@/>C&IC2X/Ddf#%Cio4Nb"!#%$'&)(*$,+.-0/.-1$'2 &)$a"B"[email protected]"CNb`N1,@.M,C%B"@_"C #N1-1XB"C4C=/yNb Nb,@9M'C%B5@;KWl CN1N0C4C%+9/y!M:,N1C B«&IÜA-1&B5& "C N1,@.M'CB"@_&)$A"B5Nb,[email protected](E,@9&I-*/>$'B"+N0C%+DM'@.,(""C+?#%,@>#/>[email protected]+9/>-0XB"C%+?,("(",@>,-0+.+.,2:/E,2"+Nb4"!#%$'&)(H$'+.-7/>-1$,2`&I$a"B"N1,-1@9C,U

åÈAÉkÉkÈ3 (;(;/@CIG = (VG, EG)

D]IH = (VH , EH)

FHD]Ke?Y?[:]\+^+DNLQc;/@CIk ∈ N∗ D]I

V ′ ⊆ VGKH1kR>/QF?KHBEDQcW;/@CI

v ∈ V ′ c BE/[L (V ′ \ v) ∪ VH

DNL]I KX1kR>/QF?KXBED

GHv

cåÈAÉkÉkÈ ;( (9;/@CI

G = (VG, EG)D]IH = (VH , EH)

FHD]Ke?'Y?[:]\+^+DNL D]Iv ∈ VG

c BE/[Lmod-wd(GH

v ) ≥ max(mod-wd(G),mod-wd(H))c

]>/1`L]I[:?I*@/1mctA$'-7/k = max(mod-wd(G),mod-wd(H))

U;lN1,-1@9C%&)C%2:/cmod-wd(G H

v ) ≥ kc#@

GC/H+.$'2:/?"C+?+9$'B"+ö6YM'@.,(""C+"C

GHv

UtA$'-0C%2:/

TG

C/TH

"C+ ,@[email protected]%+-12""B5-1+>2X/"C%+45!%#%$,&I(H$'+.-7/>-0$'2"+y(E,@k6Y&)$A"B5N1C%+y"C

GC/[email protected]+.(HC%#=/>-0h,C%&)C%2:/U ,@N1CN0C%&)&IC$x5U 5cºNV[Q,@93"@.C

T$'35/.C%2XBÄC%2 +.B"35+9/>-7/>BE2X/

v(E,@

TH

E,25+TG

C%+/B"2L,@[email protected]"5B"-1+.,2:/DB"2"Cy"!%#$'&)(*$'+9-0/.-1$'2`(",@k6Y&)$a"B"N1C+e"C

GHv

UÃ2$,35/>-0C%2:/ "$'25#uB"2"Cu!%M',N1-7/>!^(H$'B"@ NbkN1,@.M'CB"@ &)$a"B"[email protected]å+.-1&)-1N1,-1@9CPo #%C%N0N1CÆ5$'2"2"!C ,B/>5!%$'@ %&)C$naUTn,R4(H$'B"@DN1Nb,@9M'C%B"@D5Cy#N1-0aB5C,U

: Z¾ÜHÂfÂfÀ"f¾°È () ( ;/@CIG

KH1 Y?[:]\+^+DQcyBE/[Lmod-wd(G) = maxmod-wd(H) : H

DNL]I KX1Y?[:]\+^+D [D\m[NLD]1+I:?I@ <iFWV KH113/D]K!F@C1+ID][a13D$FM:?1`L B :JFHbaN/>\+/L]@CI@/1 >/QF?KHB :?@C[D FHD

G c

KC("N1B5+%c"#$'&I&)CN1Nb,@9M'C%B"@&I$a"B"N1,-1@9CKC%+/?B"2(E,@>& =/>@.C_+9-1&)-1N1,-1@9C-o4N1Nb,@9M'C%B"@"C_#%N0-1XB"C,cNbNb,@9M'C%B5@y&)$a"B"[email protected]+9/43H$'@92"!%CI+.-áC/+9C%B"N0C%&)C%2:/4+9-G#wEaB5C`M'@>(""C #%,@>#/>[email protected]+9/>-0XB"C E2"+N1

k ! 7 / !

"!#%$'&)(*$,+.-0/.-1$'2 C2 #%$,B"(*C+LC+9/^"C«N1,@.M,C%B"@ &)$A5B"Nb,[email protected]Æ3H$'@92"!%CUWYN_C2 C%+/^"CÄ&)%&)CÄ(H$'B"@N1"!#%$'&)(*$,+.-0/.-1$'2I3"-76789$'-12:/%UaÃ_2 (HC%BA/D&I$,2X/[email protected]@("N0B"+("@9!%#-1+.!&IC2:/aB5CE,2"+N1CK#%,+e"B 35-76089$,-12:/$,2 Nb@9C%N1/>-0$'2 +.B"-7h,2:/>C,U"$# ÌXZ¾%aÉkÈ3 ;())(-;/@CI

GKH1 Y?[:]\+^+DQcWBE/[NL

cwd(G) ≤ 4× maxcwd(H) : HDNL]IBED_Y?[7:]\+^+D

a7:?[:>a]I][a@EL]I@BAQKDiFWV KX113/D]K`F@C1+ID][]13D$FHD_B :GFHbaN/>\+/L]@CI*@/1 OQ@CRTSU/@C1+IyFHDG

c

]>/1`L]I[:?I*@/1mc?Ã2ý&)$'2:/>@9C aB5Cmod-wd(G) ≤ 2 × mod-wd(G

v)C/

mod-wd(Gv) ≤ 2 ×

mod-wd(G)U

y¾ÜZÇHÎr°Á* (;/@CIG = (V,E)

KH1PY?[:]\+^+DD]IW ⊆ V

c ;/@CIV ′ KH1 k R>/QF?KHBED$FHD G c BE/[NL V ′

DNL]I KX12kR>/QF?KXBEDiFHD

GW

]>/1`L]I[:?I*@/1mctA$'-7/CB"25C4#%Nb+.+.C45C

V ′ E,25+ G U KN0$'@.+ C ∩W C+9/-02"#%N0B"+E,25+B"2"C #%N1,+.+9C"CV ′ E2"+ GW cEC/

C \WC%+9/?-02"#%N0B"+DE,2"+B"25Cy#Nb,+9+.C"C

GW U

Ã2L$'3A/>-1C2:/"$'2"#-1&)&)!%"-1/>C&IC2X/y¾ÜZÇHÎr°Á* ( ;/@CI

G = (V,E)KH1 Y?[:]\+^+D D]I

W ⊆ Vc @

FDNL]IKH13D FHbaN/>\+/L]@CI@/1

kR

>/QF?KHB :?@C[NDiFHDGy:?BE/[NL

FDNL]I KX13DiFHbaN/>\+/L]@CI*@/1

2kR>/QF?KHB :?@C[D FHD

GW c

y¾ÜZÇHÎr°Á* (y;/@CIG = (V,E)

KX1Y?[:]\+^+D D]Iv ∈ V

cmBE/[Lmod-wd(G) ≤ 2×mod-wd(G

v)c

]>/1`L]I[:?I*@/1mctA$'-7/kNbyNb,@9M'C%B"@å&)$A5B"Nb,[email protected]"C

Gv

= (V \ v, E ′)C/FB52"CK"!#%$'&)(H$'+.-7/>-1$,2

k6g&I$a"B"N1,-1@9C"C

Gv UWtA$,-0/

F ′ = F ∪ V U KN1$,@.+

F ′ C%+/B"25CI"!#%$'&)(H$'+.-7/>-1$,2 k 6Y&)$a"B"[email protected] "CG′ = (V,E ′)

C/?"$'2"#B"25Cy5!%#%$,&I(H$'+.-7/>-0$'22k6g&I$a"B"N1,-1@9C5C

G = G′NG(v) Uy¾ÜZÇHÎr°Á* (y;/@CI

G = (V,E)KX1Y?[:]\+^+D D]I

v ∈ VcmBE/[L

mod-wd(Gv) ≤ 2×mod-wd(G)

c

]>/1`L]I[:?I*@/1mctA$'-7/kN1 Nb,@9M'C%B5@K&)$A"B5Nb,[email protected]

k"C

GC/FB"2"C4"!#%$'&)(*$,+.-0/.-1$'2


GUtA$'-7/

F ′ = X \ v : X ∈ FU!N1$'@9+

F ′ C%+/B52"C"!%#$'&I(H$'+9-0/>-0$'2 k 6Y&)$A"B5Nb,[email protected]?"C G c"C=/"$'25#yB52"C"!%#$'&I(H$'+9-0/>-0$'22k6Y&)$A"B5Nb,[email protected]?"C

Gv

= G− vNG(v) U

N1,@.M'CB"@_&)$A"B5Nb,[email protected]%+9/y"$,2"#),B«(5N1B"+"CBA dÁ$'-1+KNb Nb,@9M'C%B5@_&I$a"B"N1,-1@9C"C

Gv cøC=/"$'25#IB«("N0B"+_"CBA dÁ$'-1+_N0C4& °5-0&B"& "C)Nb`[email protected]'CB"@_&I$a"B"N1,-1@9C"C+M'@>,(5"C%+y#,@.,#/.!%@9-1+9/.-1XB"C+E,25+?N14"!%#$'&I(H$'+9-0/>-0$'2 &)$A"B5Nb,[email protected]

Gv UKC("N1B"+cE+.-

HC%+/KB"2M'@.,(""C#@>,#=/>!%@9-1+/>-1XB"C"CN1"!#%$'&)(*$,+.-0/.-1$'2&)$A5B"Nb,[email protected]_"C

Gv cH,N0$'@.+-1NC 5-0+9/.C B"2^M'@>,(5"C #%,@>#/>[email protected]+9/>-0XB"C

H ′ "CyNb)"!%#$'& 6(H$'+.-7/>-1$,235-76089$,-12:/D"CGcEC/B"2

w ∈ V (H ′)/>C%N0+aB5C

H = H ′wUmK$'2"#[email protected]'CB"@D&I$a"B"N1,-1@9CC%+/,B ("N0B"+

4dÁ$,-1+N0C &IÜA-1&B5& 5C%[email protected]'CB"@.+&)$a"B"[email protected]+"C%+4M'@.,(""C+4#,@.,#/.!%@9-1+9/.-1XB"C+4E,2"+N1"!#%$'&)(*$,+.-0/.-1$'2 35-76089$,-12:/e"C

GU

u

j7d npoNkt^|kGfV_z Kx fô :II oNkt^|kGfg ow' "!#%$'&)(*$,+.-0/.-1$'2

k6Y&)$A5B"Nb,[email protected],#%$,&I&)CNb_"!%#$'&)(*$'+9-0/.-1$'2 C%2I#%N0-1XB"C%+áC/áN1y"!#%$'&)(H$'+.-7/>-1$,2

;KWlKcA+.$'2:/D3E+.!%C+D+.B"@"C%[email protected]"@9C%+eC2"@>,#-12"!+%U5CK(E,@>& =/>@.C?+.-0&I-0Nb,[email protected]/.@.$a"B"-7/G@9!%#C%&)&IC2X/%[email protected]'CB"@D"C@>,25M"c"C%+/?aB",2"o)N0B"-º3E+.!y+.B"@B"2L,@[email protected]_2"$'2C%25@>,#-12"!UÃ2-02:/>@.$a"B"-0@>",2"+#%C=/./>C+9C%#/.-1$'2N1"!%#$'&I(H$'+9-0/>-0$'2k6Y3"-0&I$a"B"N1,-1@9C,cXXB"-C%+/?B"2"CM'!%25!%@>N1-76+>Ü/>-1$,225$'2C%2"@.,#%-02"!"CNb"!#%$'&)(*$,+.-0/.-1$'2

k6Y&)$A5B"Nb,[email protected]

4 . ,& <$^2 Ì ÁmfH ;( ()+(tA$'-7/

k ∈ NC/G = (V,E)

B"2 M'@.,(""CUA?2"C3"-Å6Y(E@9/>-7/>-0$'2V1, V2

"CVC+9/B"2

k.<fÉkøËWpÂfÈ ; "C

G+.-V1

C=/V2

+.$'2:/?"C+k6Y&)$A"B5N1C%+"C

GU

øC%+16Y3"-0&I$a"B"N0C%++.$'2:/åN0C%+Z89$,-12:/>B"@9C%+C/N0C%+<#$6089$,-12:/>B"@9C%+cC=/N1C+

26g3"-1&)$A5B"N1C++9$'2:/"C+#%$'B"(HC%+c"C+#$6Y#$'B"(HC%+$'B"C%+?3"-Å6089$'-02X/%UÃ2 h,C%@9@>E,2"+Nb +9C%#=/>-1$,2 xAUQS #%$,&I&)C%2:/ $'2 (HC%B5/5!%#%-0"C%@C2 /.C%&)("+(*$,N<a2"$'&)-b,N+9-_B"2M'@.,(""Cá(*$,+.+7 "CeB"2

k6Y35-1&)$A"B5N1Cü2"$,2/>@9-0ha-b,NVc%(H$'B"@

kO5A!,U,Ã2h'C@.@.!%M',N1C&IC2X/XB[]B"2"Ce+/>@.B5#/>[email protected]@.C("@.!+.C2X/>,2:/D/>$'B5+eN1C+

k6g3"-1&)$a"B"N1C+å"B M'@>(""CK(HC%BA/D/[email protected]#%,N1#B"N1!CC2`/>C%&)("+e(H$'N7A25$'&I-1,Nôc(*$,B"@

kO5A!,U[Q,@93"@.C[]-12"#N1B"+9-1$'2)"C%+GC%2"+9C%&35N1C%+GdÁ$'@9/.+G[]B"2"C?df,&)-1N1N0Ce(E,@/>-7/>-0h,C25$'B"+5$'2"2"CB"2 &)$H'C%2 "C+9/.$A# KC%@C2k/w-1N1N0CN1-02"[email protected]/.$'B"+ N1C%+yC%25+.C%&3"N1C+"CINb df,&I-0N1N0C,U3-12"+9-åNô[\,@.3"@9CI5C "!#%$'&)(*$,+.-0/.-1$'2&)$A"B5Nb,[email protected])C%+9/ B52"C @.C("@.!+.C%2:/>/>-0$'2 #%$'&)(E,#=/>C "CI/>$'B"+4N0C%+&)$a"B"N1C+[]B"2M'@.,(""Cc,N1$,@.+XB[]-1N(HC%B5/-0N ^C%2 h'$'-0@B"2 2"$'&3"@.CC A(*$'25C%2:/>-0C%N Á(E,@C 5C&I(5N1C(H$'B"@B52ÄM,@>,("5C#%$'&)("N0C/cH/>$'B5+N1C+C%25+.C%&3"N1C+ +.$,2X/I"C+I&)$A5B"N1C+ =Uå[\,@[email protected]"CL5!%#%$,&I(H$'+.-7/>-0$'2 C%2ý#%$'B"(HC%+IC/I3"-76789$'-12:/45$'2"2"CN1C+&)%&)C%+D@.!+.B"N7/w/.+(*$'B5@DN1C%+#$'B"(HC%+C/?N1C+D3"-76789$'-12:/%U df,&I-0N1N0Cy5C%+

k6g3"-1&)$a"B"N1C+2º[\C+9/y(E,+35-76Y(",@9/.-0/>-7h'CcH&I,-0+K-1NC%+/_XBE,2" &)%&)C(*$,+.+.-03"N1C "C+9/.$A# KC%@/>$'B5+?N0C%+

k6Y3"-0&I$a"B"N0C%+eC2LC+.(E,#C (H$'N7A25$'&I-1,NôU

3$ ( ! 4 . ,/&14B $ Ì ÁmfH (;(;(K?2"C ËÌAÍ'HÉkÇHÎr°ÁH

k>VÉ«øË pÂÁÀEf¾°Èu"C

G = (V,E)C%+/4B"2"C`df&I-0N1N1C,@93*$'@9!%C

F+9B"@

V"C

k6Y3"-0&I$a"B"N0C%+%c"/.C%N0N1CyaB5CE,2"+Nô[\,@[email protected]@.C%(5@.!%+9C%2:/wÜ/>-0då5C

FcH#w",XB"C42"$aC%B"+.$,-0/"C"CM'@.!y,B("N0B"+

3U

Ì ÁmfH (;( ;(D ÂfÀ"¾zZÈHp¾VÉ«øË pÂÁÀEf¾°È "CGC+9/_N0C("N0B"+(HC/>-7/

k/.C%NaBº[\-0N<C=A-0+9/>CB"2"C"!#%$'&)(*$,+.-0/.-1$'2

k6Y3"-0&I$a"B"N1,-1@9C"C

GU

Ã2 +>,-7/XB"C#C2"$'&3"@.CC%+9//>$'BÜ89$,B"@.+e5!OE2"-VcA#%,@B52"C(E,@/>-7/>-1$,2V1, V2

C+9/e/.$'BÜ89$'B"@9+eB"2n63"-0&I$a"B"N0C,c'"$,2"#?/.$'B5/GM'@>(""C?C%+9/G"CNb,@9M'C%B"@å3"-0&I$a"B"N1,-1@9Ce,BI("N1B5+

nU5Ã2I2"$,/>C@>

bimod-wd(G)NbN1,@.M,C%B"@D3"-0&I$a"B"N1,-1@9C5CGU2 M'@>,(5"Cå+.C%@."Cá[email protected]'CB"@ø3"-1&)$a"B"[email protected]

k+.-XC/+.CB"N1C&IC2:/<+9-:-1N:C=A-0+9/>CáB"2 ,@[email protected]á2"$'2yC%2"@.,#%-02"!

T/>CN*XB[]-1NEI,-0/GB"2"C3"-¢89C#/.-1$'2)C%2:/>@9C

VC=/eN0C%+GdÁCB"-1N0N1C+á5C

TcAXB"C#wE,XB"C_2"$aC%B5`-12:/>[email protected]"C+.$'-7/á"C"CM'@.!y,B("N1B5+

3c"C=/XB"Cy(H$'B"@D#wE,XB"C ,@9/>C

e"C

TcC1

e , C2e-125"B"-1+9CB"2

k6Y3"-0&I$a"B"N0C,U

ÐºÙfÙôÖÕ%ÙVÓQâ=ÕÕ°ÖHûÜÝæaõâ=Õm÷ÅâÕÜßmÒôÖHÝ>ó=Ö.õ"×\Ö.æXÞ°ÓQØeâß°ò°×\Ö.æAß°Ö ÛàÚ9à_âÕ %û,â=ò°ÒA×QÝßmä9õâ=Øeûâ%æfÓ\ÙVÓQâ=Õåß°Ö9æAÔ=ÒgÝ û°ù°Ö.æÞ°ÓQûÜÝ ÒVÙVÓëæ9ð

+ : +

åÈAÉkÉkÈ ()H;(;/@CIG

KX1JY?[7:]\+^+DQcmBE/[L_B :9B :?[YHD]KH[ OQ@C>/QF?KHB :?@C[D FHDG

DNL]I B :9B :?[*YHD]KX[ OQ@C>/QF?KHRB :?@C[D FHD

Gc

]>/1`L]I[:?I*@/1mc[Q,@93"@.Cu-02""B"-0+>,2:/ N1k6g"!%#$'&I(H$'+9-0/>-0$'2 5C

G-12"5B"-0/ !%M:,N0C%&)C%2:/ B"25C

k6Y5!=6#%$,&I(H$'+.-7/>-0$'2`"C

GUK$'2"#

bimod-wd(G) ≤ bimod-wd(G) ≤ bimod-wd(G)U

"$# ÌXZ¾%aÉkÈ ;() ;( ;/@CIGKX1Y?[7:]\+^+DQcXBE/[L

G: KX13D B :?[YHD]KH[OQ@C>/QF?KHB :?@C[DFHD

1L]@D]I;LD]KHBED]>D]1+I

L]@G

DNL]I KX1PY?[:]\+^+DjaN/>\mBED]I /KPKX1PY?[:]\+^+D LQ:?1`L :?[]IDNLQc]>/1`L]I[:?I*@/1mctAB"("(H$'+9$'2"+eN0C#$'2:/>@>[email protected]$'@9+G",2"+eNV[Q,@93"@.C?"C"!%#$'&I(H$'+9-0/>-0$'2c:B"25C_,@.=/>CK,B&)$'-12"+-02""B"-7/<B"25C89$'-02X/.B"@.Cf+9-12"$,2N1CáM'@.,(""CG2[QB"@>,-7/<(",+<[\,@9/>C =cÜC/ü,B&)$'-125+B"25C#%$Ü6089$'-02X/.B"@.Cf+9-12"$,2)-0N5+9C%@>-0/å#%$,&I("N0C/ =U $'2"#N1CDM'@.,(""C2[]C%+/á(E,+å#%$'252"C=AC?C/å+.$,2I#$'&I(5N1!%&)C%2:/>,-1@9C2[]C%+/(E+#%$,2"2"C=ACcH#$'2:/>@.,"-1#=/>-0$'2UÃ2h,$'-0/?XB"CyNô[\,@.3"@9C_"Cy"!%#$'&I(H$'+9-0/>-0$'2 C%2#%$,B"(*C+º[\B"2LM'@>(""C"-1+/w,25#%C"!%@9!%"-7/w,[email protected]_2"$,B"+"$'252"C_"-1@9C%#=/>C%&)C%2:/DB52"C"!%#$'&I(H$'+9-0/>-0$'2 -02""B"-0+>,2:/DB52"C_Nb,@9M'C%B5@3"-0&I$a"B"N1,-1@9C?5C

2U5WYNºC%2 C%+/"C&)%&)C_(*$'B5@DN1C%+D#$6Y#$'B"(HC%+?C/N0C%+3"-76789$'-12:/%U

"$# ÌXZ¾%aÉkÈ ;() &(;/@CIG

KX1 Y?[:]\+^+DQcG

:GKX13DB :?[*YHD]KX[OQ@C>/QF?KXB :?@C[DAFHD2L]@D]IyLD]KXBED]>D]1+IL]@

GDNL]IKX1Y?[:]\+^+D F?@EL]I:?1KaND^+][7F?@CI:?@C[NDWBED aN/>\mBE]>D]1+I FWV KX1Y?[:]\+^+DF?@EL]I:?1KaND^+][7F?@CI:?@C[ND_/KKX1

Y?[:]\+^+D LQ:?1`LC5

I:?KH[ND7:?K YHD]> D]I(aN/RYHD]> @C1mF?KH@CIc]>/1`L]I[:?I*@/1mctA-

GC%+9/B52«M'@.,(""C"-1+/w,2"#C "!%@9!%"-7/w,[email protected],cHN1C4#$'&)("N1!&IC2X/K[]B"2«M,@>,("5C4"-1+ö6/w2"#%C5!%@.!"-0/>,-1@9C$'BuB52uM'@>,(5"C+>2"+

C5

c*/>,B"@9C,Bc*M'C%& C/#$6YM,C%& -12"5B"-0/%cº,N0$'@.+GC+9/_#$'&46("N />C&IC2:/D"!%#$'&)(*$'+.,3"N0CK(E,@GB"2"C_"C+e"!#%$'&)(*$,+.-0/.-1$'25++.B5-0h,2:/>C+ 5"!#%$'&)(H$'+.-7/>-1$,2 C%2 #$'B"(HC%+c"!#%$'&)(*$,+.-0/.-1$'2 C2 #%$Ü6Y#%$,B"(*C+%cX$'BI"!%#$'&I(H$'+9-0/>-0$'2 3"-76789$'-12:/%U ,25+G/>$,B"+GN0C%+á#,+cX+.$'-7/

TNô[\,@.3"@9C"C"!#%$'&)(*$,+.-0/.-1$'2ºUÃ2u#%$,2"+9/[email protected]"-7/_B"2Ä,@[email protected]

T ′ C%2Ä°89$,B5/w,2:/_5C%+y,@.=/>C+ o`Nô[\,@.3"@9C T (*$'B5@+.!(E,@.C@N1C+?h,$'-1+9-12"+?"C+2"$aC%B""+"C5C%M'@9!+.B5(*[email protected]"@ o3cE/w2X/XB[]-1NC%2^C A-1+9/.C,U $'BA/>C%+N1C+K,@9/.C%+K-02:/>@.$Ü6"B"-7/>C+_-12""B5-1+.C2:/_B"2"C #%$,B"(*Cc#%$Ü6Y#%$,B"(*C4$,Bu3"-76789$'-12:/cH#@/>$'B"+_N0C%+25$ACB""+E,2"+_NV[[email protected]"@9C

T+9$'2:/"!M'!%2"!@.!+%U K$'2"#Nô[\,@93"@.C

T ′ -125"B"-0/eB"2"C 26Y5!%#%$,&I(H$'+.-7/>-0$'2 "C

GU

G?C& @.XB"$'2"+XB"C +.-V1, V2

C%+/ B52"C2(",@9/.-0/>-0$'2Æ"C

GC=/yaB5C

V1

C=/V2

2"C +.$'2:/y(E,+"C+&)$A"B5N1C%+cü,N0$'@.+V1, V2

C+9/)C=A#%N0B"+.-7h'C&IC2X/)+9$'-0/4B52"C#%$'B"(HC,c<+.$'-7/ B"2"C #%$Ü6Y#%$,B"(*Cc+.$'-7/4B"23"-Å689$'-02X/%U*Ã_2Æ,("(HC%N1N0C%@.)B"25C#%$'B5(*C ("@.$'([email protected]"C#$'B"(HCXB"-2º[\C+9/_2"-B"2"C#%$6g#%$'B"(HC,cH2"-B"2Æ3"-Å6089$'-02:/UKC&)%&)C_B"2"C#%$6g#%$'B5(*C("@9$'("@.C+9C%@> B"2"C#$6Y#$'B"(HCyXB"-ø2º[\C+9/?2"-øB"2"C#$'B"(HC,cZ2"-B"23"-Å6089$'-02X/%c5C=/B"23"-76789$'-12:/("@.$'([email protected]%+/B52L3"-Å6089$'-02:/DXB"-ø2[]C%+9/?2"-øB52"C#%$'B"(HC,cE25-øB"2"C#%$6g#%$'B5(*CU

tAB"("(H$'+9$'2"+XB"CGC%+/B"2«M'@>(""C)"CIN1,@.M,C%B"@_3"-0&I$a"B"N1,-1@9C

2aB5-G2[]C%+/(",+ ",2"+B52"CI5C%+#%N1,+.+9C%+#%-7/>!C%+%U

le,+9WYNGC 5-0+9/.C`B"2"C ,@.=/>Ce5C

TXB"--02""B"-7/yB"2 3"-Å6089$'-02X/y("@9$'("@9C,U l$'&I&)C

G2[]C%+/(E+B"2M'@>,(5"CÆ+>2"+

C5

cå/w,B"@9C,BºcM'C& C=/`#%$Ü6YM'C& -02""B"-7/cá,N1$,@.+)-1N?C=A-1+/>CÆB"25Cu,@.=/>CfXB"-2[\-02""B"-7/(E,+B"23"-76789$'-12:/%U

le,+'U0gf-12""B5-0/ B"25CÄ#$'B"(HC ("@.$,("@.CUtA,2"+(*C@9/.CÄ5CÄM,!%2"!@>,N0-0/>!c

C1e ⊆ C1

f

UtA$'-0/V1 = C1

e

cV2 = C2

e \ C2f

C=/V3 = C2

f

UKN0$'@.+V1 ∪ V2, V3

C+9/ B"2"C)#%$'B"(HC("@9$'("@.Ccº"$,2"#-0NC A-1+9/.Cv ∈ V3

/>C%NXB"CN(v) ⊆ V3

c*C=/V1, V2 ∪ V3

C%+/_B"23"-Å6089$'-02X/("@.$,("@.CcE"$'2"#/.$'B5/+.$,&I&)C/D5CV1 ∪ V2

,B&I$,-12"+eB52h,$'-1+9-12 E2"+V3Ul$'2:/>@.,"-1#=/>-0$'2U

u

le,+'UTnGf-12""B5-0/DB"25Cy#$6Y#$'B"(HC("@.$,("@.CUml$'&I(5N1!%&)C%2:/>,-1@9CK"B#,+'U7'U

le,+-nJWYNá2º[\C 5-0+9/.CIB"25C`,@9/>Ce5C

TaB5-á-02""B"-7/B"2k35-76089$,-12:/("@.$,("@.CUWl$'&I&)C

G2[]C%+9/2"-åB"2M'@>,(5"C^"-0+9/>,2"#%C"!@.!%5-0/[email protected]å2"-N1C#%$'&)("N1!&IC2:/Iº[\B"2 M'@>,(5"CL5!%@.!"-0/>,-1@9C,cå-1NDC A-1+9/.C"C%BA ,@.=/>C+

eC/f/>CNXB"C

e-125"B"-0/ B"2"C #%$'B"(HC`("@9$'("@9C`C/

f-12"5B"-1+9C B"2"C #%$6g#%$'B"(HC("@.$'([email protected],U`t5,25+e(*C@9/.CK"CM'!%2"!@>,N0-0/.!,c

C1e ⊆ C1

f

U!ta$'-0/V1 = C1

e

cV2 = C2

e \C2f

C=/V3 = C2

f

UKN1$,@.+

V1, V2∪V3C%+9/üB"2"CD#%$,B"(*C("@.$'([email protected],c"$'2"#-1NXC=A-1+/>C

u ∈ V1

/.C%N5XB"CN(u) ⊆ V1

c,C=/V1∪V2, V3

C+9/B"2"C?#%$6g#%$'B"(HC?("@.$'([email protected],ca"$'25#-0NEC=A-1+/>Cv ∈ V3

/>CN*XB"CV1 ∪V2 ⊆ N(v)

Ul$'2:/>@>"-1#=/>-1$,2U

l$'2:/>@.,-1@9C%&)C%2:/goÆNb^N1,@.M,C%B"@&)$a"B"[email protected]$'2ÏÆ"$'2"# B"25C #,@.,#/.!%@.-0+>/.-1$'2 5C%+4M'@.,(""C+4"[email protected]'CB"@D3"-1&)$a"B"[email protected],B("N0B"+2UZÃ2(*CB5/[email protected]& ,@9XB"C%@XB"CN1C

C5

2[\C2df-0/(E,+D(",@9/.-1C,U $ 684<,/9 $+ #$ 1 . ,& (4 $'$ 6 #$ ( &($ 9 "$# ÌXZ¾%aÉkÈ ;(*)>;;(-;/@CI

GKH1PY?[7:]\+^+DQc3BE/[NL

mod-wd(G) ≤ bimod-wd(G) ≤ 2mod-wd(G) c]>/1`L]I[:?I*@/1mctA$'-7/

F ′ B"2"C^"!%#$'&I(H$'+9-0/>-0$'2 k 6Y35-1&)$A"B5Nb,[email protected] "C G cáC/ v ∈ VU KN0$'@.+

F =V ′ : v 6∈ V ′ C=/ V, V \ V ′ ∈ F

∪ V, v

C%+/B52"C"!%#$'&I(H$'+9-0/>-0$'2k6g&I$a"B"N1,-1@9C5C D4U

tA$'-7/FB"25CI"!#%$'&)(H$'+.-7/>-1$,2

k6Y&)$A5B"Nb,[email protected]

GUKN1$,@.+

F ′ =V ′, V \ V ′ : V ∈ F

C%+/B"2"C"!#%$'&)(H$'+.-7/>-1$,22k6g3"-1&)$a"B"[email protected]?"C

G(",@N1C_N1C&I&)C_x5U\S5U

: Z¾ÜHÂfÂfÀ"f¾°È ;()< (;/@CIG

KH1 Y?[:]\+^+DQc3BE/[LNLC-wd(G) ≤ cwd(G) ≤ 2× bimod-wd(G)

D]Ibimod-wd(G) ≤ 2NLC-wd(G) ≤ 2cwd(G) c

"$# ÌXZ¾%aÉkÈ ;(*)? (-;/@CIG

KH1PY?[7:]\+^+DQc3BE/[NLrwd(G) ≤ bimod-wd(G) ≤ 2rwd(G) c

]>/1`L]I[:?I*@/1mc[Q,@93"@.C -02""B"-0+>,2:/Nbk6Y"!#%$'&)(H$'+.-7/>-1$,2u"C

G-12"5B"-0/!M:,N1C&IC2:/B"2"C Nb,@9M'C%B5@"C@>2"M "C

kcE#@DB"2

k6Y3"-0&I$a"B"N0C B"2@>,2"M)[\,B("N1B5+

kU

tA$'-7/B"2"Cá3"-76g(E,@9/.-0/.-1$'2V1, V2

"Cå@>2"MlUtA$,-0/

V2 = v1, . . . , v|V2|UN1$'@9+ø-1N'C A-1+9/.C

lh'C%#=/>CB"@.+"C

U1, . . . , Ul

"C/w,-0N1N0C|V2|

/.C%N1+IXB"CÆ(H$'B"@I/.$'B5/u ∈ V1

cGN0CLh'C#/.C%B"@(a1, . . . , a|V2|)

C+9/`B52"C#%$,&3"-12",-1+9$'2 N0-12"!%,-1@9C?5C%+Ui

cA$ai = 1

+9-vi ∈ N(u)

c0+9-12"$,2U5WYN_`B`("N1B5+

2l#$'&3"-02E,-0+.$'2"+N1-02"[email protected]+_"C%+

Ui

/.C%NåXB"C)/.$'B5/.C%+N0C%+("@9$Ü89C%#=/>-1$,2"+ 5C%+#$'&3"-02E,-0+.$'2"+_+9$'-1C2:/E2"+0, 1

UøWYN Ä"$'25# B(5N1B"+2l,Ü8.,#C%2"#C%+?(*$'+9+.-03"N1C+D[\B52^+.$,&I&)C/D5C

V1

E,2"+V2

UKC &)%&)C,c-0Nk,B (5N1B"+

2l,°8.,#%C2"#%C+(*$,+.+.-03"N1C+ 5C

V2

E,25+V1

U(E,@/>-0/.-1$'2V1, V2C%+/"$'2"#uB52"C

2l(E,@9/.-0/.-1$'2Uá?2½@.3"@9CÆ-12"5B"-1+.,2:/ B"2"Cu"!#%$'&)(H$'+.-7/>-1$,2 "Cu@.,2"M

l-12""B5-0/`B52"C

2l6Y5!%#%$,&I(H$'+.-7/>-0$'2U

jFq4 xoÁxoVt vz

Ã2`("@.!+.C2X/.CE,2"+G#%C/9/>C+9C%#=/>-1$,2 B"2`&)$O,C%2 "C/>@.$,B5h'C@G/>$,B"+eN1C+k6Y&)$A"B5N1C%+áC/

k6Y35-1&)$A"B5N1C%+[]B"2 M,@>,("5C?C2)/.C%&)("+á(*$'N7a2"$'&)-b,NVcÜ(H$'B"@åB"2

kO5A!,UHKC?("N1B5+%c:$'2 h'C@.@>yXB"C?/.$'B"+åN1C+

k6Y&)$A5B"N1C+C/

k6Y3"-0&I$a"B"N0C%+(HC%BAh'C%2:/K/>@9C+9/>$a# K,!+C2LC+.(E#%Cy(*$'N7a2"$'&)-b,N$f(H$'B"@

kO5A! U

F

*j9(4<684<,/9 Ì ÁmfH ;( ()+(_ta$'-0/

F = (X1, X2, EF )B"2 M,@>,("5C35-1(E,@/>-eC=/

G = (V,E)B"2 M'@.,(""CU2Ä#%$,B"("N1C

(V1, V2)/.C%NXB"C

V1, V2+9$'-0/B52"C 3"-76g(E,@9/.-0/.-1$'2L"C

VC+9/yB"2"C

F>ÇÀ"¾m°r°fHÏ"C

G+%[]-1NGC=A-1+/>C B"25C`dÁ$'2"#/.-1$'2f : V → X1 ∪ X2

/>CN1N0C XB"C∀i ∈ 1, 2

c∀v ∈ Vi

cf(v) ∈ Xi

cüC=/∀(u, v) ∈ (V1, V2)

cu, v ∈ E ⇐⇒ f(u), f(v) ∈ EF

UÃ2 h'$'-7/df,#-1N1C&IC2:/XB[]-1NG+%[\,M'-0/ []B"2"C M'!2"!%@.,N1-0+>/.-1$'2Ä"B &)$A5B"N1C f(H$'B"@

F = F1

=c"C`N1#%$,B"(*C Á(*$'B5@F = F2

c"C/?"B3"-Å6089$'-02X/ f(H$'B"@F = F3

=U

F1 F2 F3Ã2«h'$,-0/ aB5CV1, V2

C+9/B"2k6g3"-1&)$a"B"N1C+.-åC/+.C%B5N1C%&)C%2:/ +.-å-1NC A-1+9/.C B"2«M'@>(""C)3"-1(E@9/>-

F = (X, Y, E)h'C#

|X| ≤ kC=/|Y | ≤ k

cA/.C%NXB"C(V1, V2)

+.$'-7/DB"2"CF6Y(E@9/>-7/>-0$'2UHKC_&)%&)C,c

V ′C%+/KB52k6g&I$a"B"N0C+.-C/+.CB"N1C&IC2X/+.--0NøC=A-1+/>CyB"2^M'@.,(""Cy3"-0(E,@9/.-

F = (X, Y, E)h,C%#

|X| ≤ kC/|Y | ≤ 2k

c5/>CNøaB5C(V ′, V \ V ′)

+.$'-7/B"2"CF6g(E,@/>-0/.-1$'2ºU

,W476" . $ 01, (T68 , $ 19$F 0 # 684<684<,/9

Ã2 ("@.!+.C%2:/.C B"2,N1M,$'@.-7/>"&)C (*$'N7a2"$'&)-b,N(H$'B"@y/>@.$,B5h'C@4N1C%+F6g(E,@/>-0/.-1$'25+yº[\B"2 M,@>,("5C

Gch'C#

F6gOA5!UøCu2"$'&3"@9Cu"C%+

F6Y(E@9/>-7/>-0$'2"+ (HC%BA//>@9CuC=A(H$'2"C%2:/.-1C%N_C2½N1k/w,-0N1N0CL"C

Gce&I,-0+$'2 h,C%@9@>kXB"C^/.$'B5/.C%+`N0C%+

F6g(E,@9/.-0/.-1$'2"+ (HC%B5h,C%2://[email protected]+9/>$a# K,!C%+ E2"+`B"2"CL+9/[email protected]"#=/>B"@9CÆ"C^/>,-1N0N1C(H$'N<a2"$'&)-bN1C,U

y¾ÜZÇHÎr°Á* ;() ;($;/@CIF = (X, Y, EF )

ID]B AQK V @CBD@? @EL]IDx ∈ X

D]Ix′ ∈ X

x 6= x′

AD]IN(x) = N(x′)

cBE/[L(V1, V2)

DNL]IKH13DFRE\`:?[aI*@CI@/1 FHD

GL]@AD]I LD]KXBED]>D]1+I L]@D]BCBED DNL]I$KX13D

(F − x)RE\`:?[]I@CI@/1 FHD

Gc

K$'2"#c5+.,2"+e(HC%@/>C"CKM'!%2"!@>,N0-0/.!,cX$'2`+9CN1-0&I-7/>C%@.,BAIM'@>(""C%+3"-1(E@9/>-0+á/.C%N0+XB"CN0Ch'$,-1+.-02E,M'C"C"CBA+.$'&)&)C/>+D5-*!%@.C2:/>+?,("(E@9/>C2E,2:/-o4Nb&)%&)C_#%Nb+.+.C+9$'-0/D/.$'BÜ89$'B"@9+"-*!%@9C%2:/U Ì ÁmfH ;( ;(E;( tA$'-0/

F = (X, Y, EF )UHÃ2L"-0/XB"C

(V1, V2)C%+9/B"2"C

F>ÇÀ"¾m°r°Á*ýÇ¾°ZÇ¾°È"C

G+9-

(V1, V2)C+9/KB"2"C

F6Y(E@9/>-7/>-0$'2"C

GC/K(*$'B5@/>$,B5/

u ∈ X ∪ YcV1, V2

2[]C%+/_(E,+B"2"CF − x

UtA-

(V1, V2)C+9/B52"C

F6Y(E@9/>-7/>-0$'2cE,N0$'@.+-0NC A-1+9/.C /.$'BÜ89$'B5@.+KB52^+.$,B"+6gM'@>,(5"C

F ′ "C F /.C%NXB"C(V1, V2)

+.$,-0/DB"25CF ′ 6g(E,@9/.-0/.-1$'2`(5@.$'("@9C,UÃ2^O5AC

F = (X, Y, EF )B52ÆM'@>(""Cy3"-1(E@9/>-Vc"/>CNXB"C (*$'B5@?/.$'B5/

x, x′ ∈ XcN(x) 6= N(x′)C/(*$'B5@/>$'BA/

y, y′ ∈ YcN(y) 6= N(y′)

Uta$'-0/X = x1, . . . , xk

C/Y = y1, . . . , yl

cmh,C%#k = |X|

C/l = |Y |

U #wE,XB"C+9$'&)&IC=/"C

XC=/K"C

Y#%$'@[email protected]+.(H$'2"B"2"Cy#%Nb+.+.Cy"C

V1

C=/V2

(H$'B"@N1,XB"C%N0N1C$'2h#w"$'-0+.-0@DB"2L([email protected]%&)-1C@D!%N0!%&)C%2:/cEB"25C KM,@>,-02"C *UZ[\,N1M'$,@.-0/."&)CK#w"C@.#w"C@>)/.$'B5/.C%+?N1C%+F6Y(",@9/.-0/>-0$'2"+("@9$'("@.C+aB5C NV[\$'2 (HC%BA/$,35/>C2"-1@h'C# N0C%+M'@.,-12"C+eO5A!%C+%U

u

tA$'-7/x′1, . . . , x

′k, y

′1, . . . , y

′l ∈ V

cü"-*!%@9C%2:/>+45C%BA o 5C%BA*cü/>CN1+XB"C∀i ∈ 1, . . . , k, ∀j ∈

1, . . . , l, x′i, y′j ∈ E ⇐⇒ xi, yj ∈ EF

U$'B5@y/.$'B5/

i ∈ 1, . . . , kc+.$'-7/

V i1 = x′i

C/y(H$'B"@_/>$,B5/i ∈ 1, . . . , l

cø+.$'-7/V i

2 = y′i U

tA$'-7/V ′ = V \ (

⋃i∈1,...,k V

i1 ∪

⋃i∈1,...,l V

i2 )U[Q,N0M'$'@9-0/>5&IC#w"C@.#w"CoM'@>,25"-1@N1C+<#Nb,+9+.C%+V i

1

C=/V i

2

/>C%NaXB"C(⋃

i∈1,...,k Vi1 ,⋃

i∈1,...,l Vi2 )+.$,-0/D/>$,BÜ89$'B"@9+DB"2"C

F6g(E,@/>-0/.-1$'2 "C

G− V ′ UtA$'-7/

u ∈ V ′ U`ta-H$,2Ih'CB5/e@.°89$'B5/.C%@ u oyB52"CK"C+#Nb,+9+.C%+cA$,2 ,B ("N0B"+á"C%BAI(H$'+.+9-13"-0N1-7/>!%+yaB52"C(E,@9&I-ºN1C%+D#Nb,+9+.C+"CV1

C/?B"2"C(E,@9&I-ºN1C%+#Nb,+9+.C%+"CV2

Ut*[]-1NC=A-0+9/>C)B"2

i ∈ 1, . . . , l/>CNXB"C

u"-1+/>-125M'B"C "C%BaÄ+.$'&)&IC=/>+"C

V i2

c,N0$'@.+u2"CI(HC%BA/(E,+=/>@.C °89$'BA/>!E,2"+B"25C #Nb,+9+.C"C

V1

U!KC &)%&)C,c5+%[]-1NøC=A-0+9/>CB"25C #Nb,+9+.C"CV1

/>CN1N1CKaB5CuC%2"-0+9/>-02"M'B"C_5C%BA+9$'&)&IC=/>+%cE$,225Cy(HC%BA/(",+N1C_("N1,#%C@DE,2"+B"2"C#Nb,+9+.C"C

V2

Ut*[]-1NEC=A-0+9/>C?B"2)+.$'&)&)C/å"C

uc:aB5-E2"C(HC%BA/G=/>@9C("N1,#%!E2"+

V12"-EE,2"+

V2ca,N1$'@9+Nô[\,N1M,$'@.-7/>"&)C+%[\,@9@./.C9"-1N2º[\C 5-0+9/.Cy(",+D"C

F6g(E,@/>-0/.-1$'2I"C

Gh'C# N1M'@>-12"C#w5$'-1+9-1C,U+tA-

u25Cy(HC%BA/=/>@9C_("Nb,#!XB"CE2"+

V1cA$'2 N0C+9B"("("@9-1&)CK"C

V ′ C/$'2 Nô[\°89$'BA/>C?E,2"+eNV[\B52"-1XB"C#%Nb+.+.CK"C V1E,2"+eN1,XB"C%N0N1C?-1N(HC%B5/?/[email protected]("N1,#%!UZÃ2df,-7/D"C&)%&)C_+%[]-1N2"C(HC%B5/?=/>@.C("N1,#%!_XB"CyE,25+

V2

U<2X/?XB[]-1NC A-1+9/.C_"C%++9$'&I&)C/.+D"C

V ′ XB"-ø(HC%B5h,C%2://>@9C("Nb,#!B"2"-0XB"C%&)C%2:/?E,2"+ V1

$'BV2

c$'2N1C++.B"("([email protected]&)C_"CV ′ C=/$,2LN0C%+°89$'BA/>CE,2"+N0C%+D#%N1,+.+9C%+U

KB OE2ENôc/>$'B"+N1C%++.$'&)&IC=/>+5CV ′ (HC%B5h,C%2:/)/[email protected] ("N1,#%!++.$'-7/E2"+ V1

cü+.$'-7/E2"+V2

UÃ2+>-0/"$'25#yXB[]-1NºC=A-1+/>Cy,B&I$,-12"+eB52"CF6Y(",@9/.-0/>-0$'2`h,C%#yNbM'@.,-12"C_#w5$'-1+9-1CfC=/?,B&I$'-02"+e"CBA+9-

V ′ 6= ∅) (V1 ∪ V ′, V2)C=/

(V1, V2 ∪ V ′)+.$,2X/?"C+

F6Y(",@9/.-0/>-0$'2"+U

$'B5@/>@9$'B5h,C%@/.$'B5/.C%+KN0C%+F6g(E,@/>-0/.-1$'25+D("@.$'([email protected]%+"C

GcH-1Nø+9B /[]C%+9+> 'C@/>$'B5/.C%+N1C%+?M'@.,-12"C+(H$'+.+9-13"N0C%+

-0N5 C2^

Θ(nk+l)U

"$# ÌXZ¾%aÉkÈ> ;() ;(;/@CIF = (X, Y, Ef)

KH1GY?[:]\+^+D OQ@\`:?[aI@:?b?Dba|X| = k

D]I|Y | = l

c 1g\+D]KHIFHba]@FHD][L]@KH1PY?[:]\+^+D

G\+/LNL 7FHD KX13D

FRE\`:?[aI*@CI@/1\m[/N\m[DD]1ID]>\+L

O(nk+l+2)c

tA$'-7/k ≥ 2

OA5!UÃ2 +.,-0/ XB"C />$,B5/M,@>,("5C`(*$,+.+7 "C`B"2k6Y3"-0&I$a"B"N0CG(H$'B"@y/>$'BA/

v ∈ VcN1C+

v, V \ v +.$,2X/5C%+

k6g3"-1&)$a"B"N1C+%U tA$'-7/

rB"2 C%2:/>-0C%@KO5A!Uå[\,N0M'$'@.-7/>"&)C(*$,B"@K/>@9$'B5h,C%@B"2"C

F6Y(E,@/>-7/>-1$,24("@.$'([email protected]"$'B"+á"$'252"[email protected]#/.C%&)C%2:/B52`,N0M'$'@.-7/>"&)C(H$'N7A25$'&I-1,N5(H$'B"@/.@.$'B5h,C%@GB"2

k6g3"-1&)$a"B"N1C?/>CNøaB5C #wEaB5C #Nb,+9+.C+.$'-7/"C_/w,-0N1N0C,B&I$,-12"+

r -1N+9B /?[\C+.+.H,C%@(H$'B"@e/.$'B"+N1C+M'@.,(""C+?35-1(E,@/>-0+Dh'C%# ,B("N1B5+

max(k, r)+9$'&I&)C/.+DE,2"+B"25Cy#Nb,+9+.CU

: Z¾ÜHÂfÂfÀ"f¾°È ;( (i;/@CIk, r ∈ N∗ ?XNLQc 1 \+D]KHI_FHba]@FHD][JD]1 ID]>\+L

O(n2×max(k,r)+2)L]@KH1

Y?[:]\+^+D \+/LNL 7FHD KH1kR OQ@C>/QF?KHBED_ID]B AQKD aN^`:[AQK+Dja]B :LNLD L/@CIyFHD I:?@CBCBED$:?K>/@C1`L

rc

KCy&I&ICc5$'2$'35/.-1C%2:/ : Z¾ÜHÂfÂfÀ"f¾°È ;(E )+(-;/@CI

k, r, l ∈ N∗ ?XNLQc 1\+D]KHI FHba]@FHD][D]1 ID]>\+LO(nmax(k,r)+max(2k ,l)+2)L]@KH1PY?[7:]\+^+Dy\+/LNL 7FHD_KH1

kR>/QF?KHBED FHDiI:?@CBCBEDi:?K>/@C1`L

rD]IyFHD I:?@CBCBED$:?KA\mB KML

n− lc

Ã2L+[\-02X/.!%@9C%+.+9C &I,-02X/.C%2E2X/?BA("@9$'("@.-0!/.!%+"C%+F6Y(E@9/>-7/>-0$'2"+º[\B"2M'@.,(""CU

$'B5@</>$'BA/u ∈ V ′ cÜ+.$,-0/ f1(u)

Nô[]B"2"-0XB"Ci/>CNAXB"C

u(HC%BA/ü/[email protected]("Nb,#!E2"+

V i1

C/f2(u)

Nô[]B"2"-0XB"Ci/>C%NåXB"C

u(*CB5/=/>@9CI("N1,#%! E2"+

V i2

UÃ_2Ä+.CI@>,&g %25CoB"2k(5@.$'3"N %&)C (*$,B"@#wE,XB"C

F

+.$,&I&)C/K"Cu"C

V ′ cº$'2u-02:/>@.$a"B"-7/KB52"Ch,@.-1,3"N0C u U* h,@9-b,35N1Cy#%$'@[email protected]%+9(*$,2"qo_N1C+9$'&)&IC=/uC%+/?("Nb,#!_E,2"+

V1

*UHÃ25!OE2"-7/?Nô[]C%2"+9C%&3"N0CC"C%+#Nb,B"+9C%+"B("@.$,3"NE &ICA

C =u⇒ v : u, v ∈ V ′ C/ ¬(xf1(u), yf2(v) ∈ EF ⇐⇒ u, v ∈ E)

.

KBÄOE2E,Nå$'2kLB"2«("@.$,3"NE &IC (E,@9/.-1#B"N1-0C%@_$Ä/>$'B5/.C%+yN1C+ #Nb,B"+9C%+ +.$'2:/ "B«/a(HC

u ⇒ vch'C#

uC/v5C%+?N1-0/9/>!@>,BA (H$'+.-7/>-0dÁ+U

åÈAÉkÉkÈ ;( ( aN^`:[AQK+D0@C1`L]I:?1Ka]@:?I@/1 FHDNL0b:?[]@:MOQBEDNLqLQ:?I@EL<=:?@ELQ:?1+ICaN/[][DNL\+/1mF KH13D

FR

\`:?[aI*@CI@/1 FHDGc

]>/1`L]I[:?I*@/1mctA$'-7/zB"2"CÄ-12"+/w,25#%-bÜ/>-1$,2 "C%+h,@.-1,3"N1C+`/>CN1N1CuXB"C

V1 = V ′1 ∪ u ∈ V ′ :

z(u)C+9/ha@>,-

C=/V2 = V ′

2 ∪ u ∈ V′ : z(u)

C+9/df,BAU

tA-NV[\-02"+9/>,2"#%-1/>-0$'2)2"C+>Ü/>-1+df,-0/G(E,+Cc"N1$'@9+$'2B"25C#%N1,B"+.C

u⇒ vE2"+

C/>CN1N0CXB"C

u+.$'-7/-125+9/w2"#%-0!o ha@>,-C/

vo df,BA*U KN1$,@.+

u ∈ V1cv ∈ V2

cºC/Nô[\,Ü8.,#C%2"#C4C%2:/>@9CuC=/v2"C@9C%+.(HC%#=/>C(E,+

Fc""$'25#

(V1, V2)2º[\C+9/?(E,+B"2"C

F6Y(E,@/>-7/>-1$,2`h'C# N1 M,@>,-02"CKO5A!C,U

tA-NV[\-02"+9/>,2"#-b/.-1$'2 +>Ü/>-1+df,-0/Ccü,N0$'@.+y(*$,B"@ /.$'B"+N0C%+

u ∈ V1C/v ∈ V2

cu, v ∈ E ⇐⇒

xf1(u), yf2(v) ∈ E(F )U+K$,2"#

(V1, V2)C+9/?B"2"C

F6Y(E,@/>-7/>-1$,2U

t5/[email protected]@.Cha-0C%2:/ o«/.@.$'B5h,C%@`B52 C%2"+9C%&3"N0C^"C^+.$'&)&IC=/>+

V ′′ "B M'@>,(5"CÆ$'@9-1C%2:/.! G′ =(V ′, A)

cHh'C#A = (u, v) : u ⇒ v ∈ C

cH/.C%NXB"C∀u ∈ V ′′, ∀v ∈ V ′ \ V ′′, (u, v) 6∈ A

U;tA$,-0/FNbdf&I-0N1N1C?"C++9$'B"+6gC%2"+9C%&35N1C%+?"C

V ′ +./>-0+9df,-0+>,2:/Nb(5@.$'("@9-1!=/>!_("@9!%#%!"C%2:/.C,UåÈAÉkÉkÈ ;( (8;:<=:?>A@CBCBED

F:BEDNL \m[/N\m[a@]INL_L]KH@Cb:?1+IDNL

Mc∅ ∈ F

V ′ ∈ F

cHci@

X, Y ∈ F:?BE/[L

X ∪ Y ∈ FD]IX ∩ Y ∈ F

c]>/1`L]I[:?I*@/1mc?Ã2`h'$'-7/e-1&)&)!%"-1/>C&IC2X/åXB"C_N0C%+GC%2"+9C%&3"N0C%+

∅C=/V ′ @9C%+9(*C#/>C2:/Nb (5@.$'("@9-1!=/>!,U

tA-XC=/[email protected]%+9(*C#/.C%2:/KN1("@9$'("@.-0!/.!,cE,N0$'@.+-1Nº2º[` (",+D[\,@.#_C2X/[email protected]

XC/V ′ \X

c"2"-ºC%2:/>@9CYC=/

V ′ \ YUWYN<2[u`"$'2"# (E,+_º[Q,@9#4C%2:/>@9C

X ∪ YC/V ′ \ (X ∪ Y )

UWYNü2[ u 2"$'2 ("N0B"+(E+yº[Q,@9#C%2:/[email protected] ∩ Y

C/V ′ ∪ (X \ Y ) ∪ (Y \X)

U+K$,2"#X ∪ Y ∈ F

C/X ∩ Y ∈ F

UtA$'-7/

G′ N1CM,@>,("5C$'35/>C2XBC%2y#N1#%B5Nb,2:/øN1DdÁC@.&)C/.B"@.Cü/.@>,2"+9-0/.-0h'C G cmC=/C2 #$'2:/>@.,#/>,2://.$'B"+N1C++.$,&I&)C/.+G!aB5-0h,N1C2:/>+%UXlCM'@>,(5"C,caXB"-*C+9/B"2`M'@.,(""C?"-1@9-1M'!?,# a#%N1-0XB"C,cXC%+/eB"[email protected]("@.!+.C2X/>/>-0$'2"C4/>$,B5/>C+ N0C%+F6g(E,@/>-0/.-1$'25+K"BÄM'@>(""C)h'C%#INb`M'@.,-125C4#w"$'-1+9-1C ø#wE,XB"C)+.$,B"+6gC%2"+9C%&3"N0C

V ′ "C+.$,&I&)C/.+)5CG′ /.C%NXB[\-0N2[ý,-7/),B"#B"2"CÆ@./.C^+9$'@9/>,2:/>C"C V ′ E2"+ G′ #$'@.@9C%+9(*$'25 oÄB52"C

F6Y(E,@/>-7/>-1$,2`"-*!%@9C%2:/>CUÃ2Ä(HC%BA/ #%,N1#B"N1C@

f1(u)C/f2(u)

(H$'B"@/>$,B"+_N1C+u ∈ V ′ C%2 />C&I(5+ O(n2)

U åB5-1+_N1C4M,@>,("5CG′ (*CB5/D/[email protected]_#$'2"+9/[email protected]"-7/DC%2 /.C%&)("+ O(n2)

U $,B"@e/>@9$'B5h'C@e/>$'BA/>C%+DN0C%+F6Y(",@9/.-0/>-0$'2"+á("@9$'("@.C+D"C

Gc-1Nå+9B / [\C+.+.H,C%@4/>$,B5/>C+N1C+M'@.,-12"C+(H$'+.+9-13"N0C%+

-1N «C%2

O(nk+l)U $'B5@/>@9$'B5h'C@/>$,B5/>C+N1C+

F6Y(E,@/>-7/>-1$,2"+ü"C

GcX-0NE+.B /e5C?/.@.$'B5h,C%@á/>$'B5/.C%+áN1C+

F ′ 6Y(",@9/.-0/>-0$'2"+("@9$'("@9C%+%ca(H$'B"@ F ′ +.$,B"+6gM'@>,(5"C"CFU $'B5@?B52

FOA5!cZ-1NK C2^4B"22"$'&3"@.CO5A!U

"$# ÌXZ¾%aÉkÈ ;(!&W( ;/@CIF = (X, Y, Ef)

KH1gY?[7:]\+^+DOQ@\`:?[aI*@c 13D L]I[aKEa]IKX[D[D\m[NLD]1+I:?1+II/KXIDNLBEDNL

FRE\`:?[aI*@CI@/1`L FHD

G\+D]KXI ]I[ND a7:?Ba]KHBENDD]1 ID]>\+L

O(n|X|+|Y |+2)c

u

jöi fgz<zü_z | X_z | ýv oV_z 4 |_z xIvzoVt vz |z<tqk kKxz

$'B5@Nb)"!%#$'&I(H$'+9-0/>-0$'2&I$a"B"N1,-1@9C,cENb)"!%#$'&)(*$'+9-0/.-1$'2C%2u#$'B"(HC%+C=/Nb)"!%#$'&)(*$'+9-0/.-1$'23"-7689$'-02X/%c,N0C%+#%N1,+.+9C%+á"CDM'@.,(""C+áC2:/>-E @.C&IC2X/á"!#%$'&)(H$'+>,35N1C%+!=/w,-0C%2:/å"[email protected]'CB"@ü"C#%N0-1XB"Ce3H$'@92"!%CcC/h,-1C2:/B"2"Cy#,@.,#/.!%@.-0+>/.-1$'2 (",@C=a/>C2"+.-0$'2"+"Cy+.$'&)&IC=/>+UEøC%+M'@>(""C%+?"!#%$'&)(*$,+>,3"N0C%+D(E@N1C+G5!%#%$,&I(H$'+.-7/>-0$'2"+k6Y&)$a"B"[email protected]/

k6Y3"-0&I$a"B"N1,-1@9C%+ü+9$'2:/e!%M',N1C&IC2X/G"CNb,@9M'C%B"@á"C#%N1-0XB"C3H$'@ö62"!C,U

,2"+#%C=/./>CG+.C#/>-0$'2c°$'2 !/>B5"-1C@>N0C%+#%N1,+.+9C%+<5CGM'@.,(""C+<"!=OE2"-0C%+<(",@#C%@/w,-02"C%+C=a/>C2"+.-0$'2"+"C+.$,&I&)C/.+%UZÃ2h'C%@9@> XB[\-0Nø+.B /?"C(HC%B[\C a/>C%25+.-1$,2"+(H$'B"@XB"CNb4#Nb,+9+.C"CM'@>(""C%+"C=ha-1C%252"C"[email protected]'CB"@D"C#%N0-1XB"C_2"$'23H$'@.25!%C,U,C,2a6J^,@9#u,2"N0-1M'2"CN(E,@.N0C^[]B"2"CuM,!%2"!@>,N0-1+>Ü/>-1$,2ý"C^Nbk"!#%$'&)(H$'+.-7/>-1$,2ýC2 #%$'B"(HC%+ XB[]-1N,("(HC%N0N1C)Nb#=6Y5!%#%$,&I(H$'+.-7/>-0$'2UWYNG(",@9/"C`Nô[]$'3"+9C%@h,Ü/>-1$,2kXB[]B"2 M'@>(""C`("@9C%&)-1C@ (E@NbL"!%#$'& 6(H$'+.-7/>-1$,2uC%2k#$'B"(HC%+2[]C%+/(E+ydÁ$'@9#%!&IC2X/y("@9C%&)-1C@_(E,@Nb`"!#%$'&)(H$'+.-7/>-1$,2u&I$a"B"N1,-1@9CJø-1Nü(HC%BA/+%[\,M'-0@[]B"2 M'@>(""C4("@9C%&)-1C%@K(*$'B5@Nb`"!#%$'&)(H$'+.-7/>-1$,2Æ&I$a"B"N1,-1@9CB"XB"C%N$'2 °89$,B5/>!4B"2 +9$'& 6&)C/B"2"-0h,C%@9+.C%N f#CXB"-C%+9/KN1Cy#,+(E,@?C=AC%&)("N0C "B^M'C%& =UHI# 6Y"!#%$'&)(*$,+.-0/.-1$'2#$'2"+.-0+9/.Co)[email protected]@.!#%B"@9+.-7h'C%&)C%2:/B"25C_"!%#$'&I(H$'+9-0/>-0$'2 &)$a"B"[email protected]"B M,@>,("5C,c5(5B"-1+eB52"C_"!%#$'&)(*$'+9-0/.-1$'2IC%2 #%$,B"(*C++.B5@?N0C%+eM,@>,("5C%+("@.C&I-0C%@9+e"CNb"!#%$'&)(H$'+.-7/>-1$,2`&I$a"B"N1,-1@9C,ca,-02"+.-5C_+.B"-7/>CG89B"+9aBº[o #%CXB"C_/>$'B5+N1C+ M,@>,("5C%+$,35/>C2aB5++.$'-0C%2:/ ("@.C&I-0C%@.+y(E,@yNb"!#%$'&)(*$,+.-0/.-1$'2Ä&)$A"B5Nb,[email protected]=/Nb"!#%$'&)(*$,+.-0/.-1$'2C%2#%$,B"(*C+%U;$'B5+e,N0N1$'2"+h,$'-1@GaB5CKNby#%N1,+.+9CK"C+eM'@>(""C%+GC%2:/>- %@9C%&)C%2:/e"!#%$'&)(H$'+>,35N1C%+G(E,@á#%C=/./>CK"!%#$'& 6(H$'+.-7/>-1$,2 2[\C+9/?(E,+"CN1,@.M'CB"@D"C#%N0-1XB"C_3H$'@.25!%C,UÃ2Ï+[\-02:/>!%@9C%+9+.CoÆ"C%+ #%N1,+.+9C%+ "C M'@.,(""C+ "!OE25-1C%+4(",@4#%C@9/w-12"C+4C=a/>C2"+.-0$'2"+ "C +.$,&I&)C/.++.-0&I(5N1C%+U!ta$'-0/

uB52 +.$'&)&IC=/e[]B"2 M'@.,(""C

G = (V,E)U"Ã2`#w"C%@9#w"Cio4°89$'B5/.C%@eB52 +.$'&)&IC=/

v,BM'@.,(""C/.C%NaB5C

v25C"-1+/>-02"M'B"C (E,+"CBAu+9$'&)&IC=/>+XB"-2[\!=/w,-0C%2:/_(",+"-0+9/>-02"M'B"!+(E,@

uU$'B"@°89$'BA/>C%@

v-0Nø+.B /"$'2"#"Cy#%$,2"2E 0/>@.CN0Ch'$'-0+.-12",M'C"C

uUEWYN-

8(H$'+.+9-13"-0N1-0/.!%+e(H$'B"@#C%NbAcE+.C%N0$'2XB"C

v+.$,-0/?,Ü8.#%C%2:/?$'B2"$'2o

ucNG(u)

$'BV \NG[u]

UøC%+ü#%,+<"[email protected]"#%$,2X/.@.!+ü+.$'2:/°NV[Q°89$,B5/<º[\B"2 hA@.,-89B"&IC%,BcÜNô[\°89$'B5/[]B"2df,Ba89B5&IC%,Bc°Nô[\°89$'B5/[]B"2+.$'&)&IC=/ (HC%2"E2X/%ceNô[\°89$'BA/ []B"2ha@.,-_,2:/>-Å6089B"&)CB C/Nô[\°89$'BA/ []B"2 df,BA,2:/.-76089B5&IC%,BU[Qm89$'B5/y[]B"2 +.$,&I&)C/y,2:/.-76Y(HC%25E,2:/C+9/ Nô[\°89$'B5/y[]B"2«+.$'&)&IC=/y,Ü8.,#C%2:/Ao />$,B"+ N0C%++.$,&I&)C/.++>B5duU 25OE2ºc5N1C+"CBA "C%@92"-1C@.+e#%,+%cAN1C%+G#,+G"!%M,!%2"!@.!%+D(5B"-1+9aB5CNô[\°89$'B5/C%+/e-125"!%(HC%2"",2:/D"B

u#w"$'-0+.-Vc"+.$'2:/?Nô[\°89$'BA/[\B52L+9$'&I&)C/e-1+.$,N1!,c5C/Nô[\°89$'BA/[\B52L+9$'&I&)C/B52"-0h,C%@.+9C%NVUÄ#Nb,+9+.C"C%+I#%$'M'@.,(""C+%cå"C+ M'@.,(""C+I"-0+9/>,2"#%C+I5!%@.!"-0/>,-1@9C%+cC/I"C%+)M'@>(""C%+)+>2"+C5

c/wB"@.C%,BcEM'C& C/#$6YM'C& -125"B"-0/.+D(*CB5h'C2:/K=/>@.C"!=OE2"-0C%+"C#%C=/./>Cy&I,2"-E @.CUÃ242"$,/>C@>K(",@TT

Nô[]C=a/.C%2"+9-1$'2 °89$,B5/ü[]B"24ha@>,-89B"&)C,BºcFT

NV[Q°89$,B5/ü[]B"2df,Ba?89B"&)CBcPVNô[\°89$'BA/[]B"2I+9$'&)&IC=/å(*C2"E,2:/c

TA(H$'B"@Nô[\°89$'BA/[]B"2)ha@.,-Z,2:/>-Å6089B"&)C,Bºc

FA(*$'B5@Nô[\°89$'B5/º[\B"2df,BA ,2:/>-Å6089B"&)CBc

CV(*$,B"@NV[Q°89$,B5/º[\B"2 +9$'&)&IC=/4,2:/>-Å6Y(HC%2"",2:/c

DVNô[\°89$'B5/[\B52 +.$'&)&IC=/"$'&)-12",2:/DC/?C%2AOE2L(",@

IVNV[Qm89$'B5/D[]B"2L+.$'&)&IC=/D-1+9$'N1!U

tA$'-7/O ⊆ TT,FT,PV,TA,FA,CV, IV,DV

U"Ã2`5!OE2"-7/D(E,@GO

Nb #Nb,+9+.C5C%+eM'@.,(""C+XB"-º(HC%BAh'C%2:/=/>@.CK$'35/>C2XB"+o(E@9/>-0@Gº[\B"2 +9$'&)&IC=/-0+.$'N0!,cAmh,C%#5C%+eC A/.C%2"+9-1$'25+D"C+.$,&I&)C/.+GE2"+OU Ã2L@.,("(HC%N1N0CXB"Cg

@u ] V m m

§GTT,FT

C+9/?Nb4#%N1,+.+9C_"C%+?#%$,M'@>,(5"C%+c§GTT,FT,PV

C%+/KN1#%N1,+.+9Cy5CyM,@>,("5C%+"-1+/w,25#%C%+"!@.!"-0/>,-1@9C%+%c§GTT,FT,CV

C+9/Nb#Nb,+9+.C"C%+M'@.,(""C+#$'&)("N1!&IC2X/>,-1@9C%+e5C%+"-1+/w,25#%C%+"!@.!"-0/>,-1@9C%+%c§GTT,FT,TA,FA

C%+9/?N1 #Nb,+9+.C"C+?M,@>,("5C%++>,25+C5c"/>,B"@9C,Bc5M'C%& C=/?#%$6gM'C%& -12""B5-0/cC/?"$,2"#+.$'2:/?"CNb,@9M'C%B5@e"C#%N0-1XB"C3*$,@.2"!C,U

"$# ÌXZ¾%aÉkÈ (E;;(y8;:Ja]B :LNLD /?OQID]1+KD \`:?[BEDNL D@? ID]1`L]@/1`L:SU/KHI FWV KH1GL/>A>D]IFH/>A@C1m:?1+IqD]I :SU/KHI FWV KH1ZL/>A>D]I\+D]1mFM:?1+I @a+V DNL]I F?@C[D9B : a]B :LNLD

GPV,DVDNL]IFHD9B :?[*YHD]KH[g>/QF?KHB :?@C[D

13/1qO]/[a13NDD]IDNL]I FHDiB :?[*YHD]KH[iFHD a]B @BAQKD_13/1qO]/[a13ND?c]>/1`L]I[:?I*@/1mctAB"("(H$'+9$'2"+?Nô[]-12:h'C@.+.CUmtA$'-7/

[email protected]'CB"@D&I$a"B"N1,-1@9CK"CN1#%Nb+.+.CU

tA$'-7/s > 2k

C/G+.$,-0/Gs = (V,E)

N1CM'@.,(""C?+.B5-0h,2:/UV(*CB5/=/>@9C(E@9/>-7/>-0$'2"2"!C%2IB"2"C?#%N0-1XB"C

U = u1, . . . , uscøC/

nC2"+.C&3"N0C%+

Pi = vi,1, . . . , vi,s-02""B"-0+>,2:/B"2«#w"C%&)-12

(vi,1, . . . , vi,s)U

lE,XB"Cul

C%+/I,Ü8.,#C%2:/JoÆ/.$'B"+)N1C%+vi,j

h,C%#i ∈ 1, . . . , s

C/j ∈ 1, . . . , l

UGs

2[\Ä(E+[\,B5/[email protected]%+?,@9/.C%+%U

" ¢¡Z£ "£¦§ øCM'@>(""CG3U

$'B5@ü/>$'B5/ncGs

C+9/",2"+Nb#%Nb+.+.CGPV,DV

UWYN"(HC%B5/=/>@9CD#%$'2"+/>@9B"-0/o(E,@9/.-1@[\B524C%2"+9C%&3"N0C+9/>,3"N0Ce"C/>,-1N0N1Csc,C%2 m89$'B5/w2X/ü+.B"##%C%+9+.-7h'C%&)C%2:/GB"24+.$,&I&)C/<"$'&)-12E,2:/C/

n+.$'&)&)C/>+<(*C2"E,2:/>+U

tA$'-7/FB"25C"!#%$'&)(*$,+.-0/.-1$'2


Gs

U;ta$'-0/Va, Vb, Vc ∈ F

h'C%#Va = Vb ∪ Vc

C=/Vb ∩ Vc = ∅

c5/.C%N0+DaB5CVa

#%$'2:/.-1C%252"CB"2Pl

c"C/?XB"CVb

C/Vc

25Cy#$'2:/>-0C%2"2"C2:/K(E+Pl

Uy¾ÜZÇHÎr°Á* (

|U \ Va| ≤ kD]I|U ∩ Va| > k

c]>/1`L]I[:?I*@/1mctA-á"C%BAk+.$,&I&)C/.+

vl,i

C/vl,j

"CPl

+.$'2:/",2"+N1L&)%&)CI#Nb,+9+.CI"CVa

c<,N0$'@.+/>$,B"+N1C%+D+9$'&)&IC=/>+um

cZmh,C%#m ∈ i, . . . , j − 1

cE+9$'2:/?E,2"+Va

cE#,@-0N1+e"-0+9/.-12"M'B5C%2:/vl,i

C/vl,j

UtA$'-7/

a ∈ 1, . . . , s/>CNGXB"C

ui 6∈ Va

UWKN0$'@.+-0Ná2º[kL(E+"CI#%Nb+.+.CI"CVa

/>CN1N1C)XB"CVa ∩

ul,1, . . . , ul,a 6= ∅C/Va ∩ ul,a+1, . . . , ul,s 6= ∅

UtAB"("(H$'+9$'2"+`XB"C

|U \ Va| > kcáC=/ +9$'-0/

a1 < a2 < . . . < ak+1

/.C%N0+IXB"CÆ(H$'B"@ />$,B5/i ∈

1, . . . , k + 1cuai6∈ Va

U;tA$'-0/a0 = 0

C=/ak+2 = s

U3N1$'@9+K(H$'B"@K/.$'B5/i, j ∈ 0, 1, . . . , k + 1

ci 6= j

c,B"#B"2½+9$'&)&IC=/ "Cuai+1, uai+2, . . . , uai+1

2[]C%+9/E2"+ Nb &)%&)C^#Nb,+9+.C "C

Va

XB"Cuai

, uai+1, . . . , uai+1U3l$'&)&IC(H$'B"@/.$'B5/

i ∈ 0, 1, . . . , kcai+1 > ai

Á2"$,/>C XB"Cak+1

(HC%BA//[email protected]!%M',Noak+2

=cVa

,B5@>,-7/+9/>@9-1#=/>C%&)C%2:/("N0B"+"Ck#%N1,+.+9C%+%U l$'2:/.@>,"-0#/.-1$'2U

u

y¾ÜZÇHÎr°Á* (i/KX[jaN^`:[AQK+Dr ∈ 1, . . . , s

Pr ∩ Va 6= ∅

c]>/1`L]I[:?I*@/1mctAB"("(H$'+9$'2"+_XB[]-1N<C A-1+9/.C

r/.C%NaB5C

Pr ∩ Va = ∅U KN1$,@.+K#w",XB"C4+.$'&)&)C/",2"+

U ∩Va

+9C%@.,-0/åE2"+GB52"CK#Nb,+9+.C"-*!%@9C%2:/>CU!l$,2X/.@>,5-1#/.-1$'2 #%,@å-1NR),B"@.,-0/áBI&)$'-125+k+1

#%N1,+.+9C%+E,25+Va

U+K$'2"#(H$'B"@#wE,XB"Cr ∈ 1, . . . , s

cPr ∩ Va 6= ∅

Uy¾ÜZÇHÎr°Á* ( =]BM`:G:?KP>/@C1`L

k + 1 Pi

AQKH@ L/1+I@C1Ka]B KML FM:?1`LVa

c]>/1`L]I[:?I*@/1mctAB"("(H$'+9$'2"+D&I,-12:/>C2E,2:/GaBº[\-0N_ ,-0/G("N0B"+e"C

k+ 1 Pi

XB"-*2"C+9$'2:/D(E,+eC2:/>-E @.C 6&)C%2:/K-02"#%N0B"+E,2"+Va

UZWYN-,B"@.,-0/5$'2"# ,B^&)$'-125+DB"2"Cy#%Nb+.+.CyE,25+Va

(*$'B5@?#w",#%B"2^"C #C%+Pi

c(H$'B"@N0C%+G+.$'&)&IC=/>+,Ü8.,#C%2:/>+B52`+.$'&)&)C/e",2"+Pi \ Va

U!l$'2:/>@.,"-1#=/>-0$'2 #%,@Va

B"@>,-7/(5N1B"+G"Ck#%N1,+.+9C%+%UZWYN-),BL(5N1B"+

k Pi

aB5-25C +9$'2:/K(E+?C2:/>-E @.C&IC2X/-125#%N1B5+DE,2"+Va

cEC/?"$,2"# -0NK),B&)$'-12"+k + 1 Pi

XB"-+.$'2:/?-12"#N1B"+D",2"+Va

U;$'B5+_,@.@9-0h,$'2"+?& ,-02:/>C%2",2:/_oIB52"C#%$,2X/.@>,5-1#/.-1$'2ºU,@K25$,/>@9C#w"$'-ÅL"C

Va

cVb

C/Vc

c*,B"#B"2Pi

2[]C%+/C2X/.-E @.C%&)C%2:/ -12"#N1B"+E2"+Vb

$,BVc

UWYNk"$'25#I("N0B"+"Ck + 1 Pi

aB5-á(HC%BAh'C%2://>@9C(E,@/>-7/>-1$,2"2"!%C+C2Pi ∩ Vb, Pi ∩ Vc

UAWYN5 4,B`&)$'-125+B"2"CK#%Nb+.+.C_",2"+Vb

(*$'B5@e#wE,#B"2"C#%C+Pi

cX(*$'B5@áN0C%+á+.$'&)&)C/>+á,°8.,#%C2X/.+ oB"2 +9$'&)&IC=/G",2"+Pi∩Vc

UHK$'2"#Vb

y,B &)$'-125+k+1

#%N1,+.+9C,Ul$'2:/>@.,"-0#/>-0$'2UÏ# 6Y"!#%$'&)(*$,+.-0/.-1$'2 "!%#$'&I(H$'+9CÄ#$'&)("NE =/>C%&)C%2:/N1 #%Nb+.+.C

GPV,DVce"$'25#ÄN1 #%Nb+.+.C "C+M'@.,(""C+L#$'&I(5NE /.C%&)C%2:/"!#%$'&)(*$,+>,3"N0C%+(E,@#C/./.C "!%#$'&I(H$'+9-0/>-0$'2 B"2"CÄNb,@9M'C%B"@ 5CÄ#N1-0aB5C2"$'2k3*$'@92"!%CUlC#%-GC%+9/(E,@9/.-1#B"N1- %@.C&IC2:/ -02:/>!%@9C%+9+>,2:/c#,@y#%C 2[\C+9/4N0CI#%,+º[Q,B5#%B"2"C "C%+4B5/>@9C"!#%$'&)(*$,+.-0/.-1$'25+e!/.B""-1!C%+E,25+#Cy&)!%&)$'[email protected],U

tA$'-7/O ⊆ TT,FT,PV,TA,FA,CV, IV,DV

U ta$'-0/G ′O

N1dÁC@.&)C/>[email protected] 5!%@.!"-0/>,-1@9C "CGOf#[\C+9/[email protected]ô[]C%2"+9C%&35N1C"C+eM'@.,(""C+DaB5-º+.$,2X/e"C%+D+9$'B"+6gM'@>(""C-02""B"-7/[]B"2M'@.,(""CE,2"+GO

=U4Nb,@9M'C%B"@D5Cy#N1-0aB5C_"CNb#%N1,+.+9CG ′O

C%+/N14Nb,@9M'C%B"@"C#%N1-0XB"C_"CGOU

tA$'-7/o ∈ TT,FT,PV,TA,FA,CV, IV,DV

U5Ã2 "-7/XB[]B"2"CK#%Nb+.+.C"CKM,@>,("5C%+C+9/+9/w3"N1C(E,@o+9-/>$'BA/>CN1C+DC=a/>C2"+.-0$'2"+

o[]B"2M'@>,(5"CE,2"+Nb#Nb,+9+.CC%+/K",2"+Nb#Nb,+9+.C,UÃ2L(HC%B5/df,[email protected]%+$'3"+9C%@h,Ü/>-1$,2"+D+.B"-7h,2:/>C%+

Î%ÈX¾m¿"À5°ÁH (McGPV,TT,FT,IV = GPV,TT FT

cHcGTT,FT,TA,FA,IV,DV = GTT FT,TA,FA

clC_XB"-ºh'CB5/?"-1@9CKXB"C_+9-$'2°89$'BA/>C_B"2 +9$'&I&)C/-1+9$'N1!oB"2M'@>,(5"C"-0+9/w2"#%C_"!@.!"-0/>,-1@9C,ca$'2)/.$'BÜ89$'B5@.+KB52^M'@.,(""C "-1+/w,2"#C"!@.!"-0/>,-1@9C,cEC/K+.-$'2u°89$,B5/>CyB"2Æ+9$'&I&)C/-1+9$'N1!$'BLB"2Æ+9$'&I&)C/"$'&)-12",2:/o4B"2L#$'M'@.,(""CcE$'2L$,35/>-0C%2:/B"2#%$'M'@.,(""CU

Î%ÈX¾m¿"À5°ÁH (;/@CIO ⊆ TT,FT,PV,TA,FA,CV, IV,DV

cMci@

PV ∈ OD]I

TA ∈ Oy:?BE/[L

G ′ODNL]IL]I:MOQBED\`:?[

DVc

Hci@PV ∈ O

D]IFA ∈ O

y:?BE/[NLG ′O

DNL]I L]I:MOQBED \`:?[DV

c4 ci@

PV ∈ OD]I

CV ∈ Oy:?BE/[L

G ′ODNL]IL]I:MOQBED \`:?[

DVc

@u ] V m m

tA-H$'2 m89$'B5/>CB"2 +9$'&)&IC=/u(*C2"E,2:/oyB"2 +9$'&)&IC=/G5B M'@.,(""Cca(5B"-1+áB"2`+9$'&I&)C/

v(*C2"E,2:/

ovc<C=/4C%2AOE2 B"2 ha@>,-Üdf,BAk,2:/.-76089B5&IC%,B

wovcü,N0$'@.+

w+.C%@.u,Ü8.,#C%2:/goL/.$'B"+N0C%++9$'&)&IC=/>+[]$'@.-0M'-125Cf"C?&I&IC+9-

wC+9/eB52 +.$,&I&)C/G,2:/>-Å6Y(HC%2"",2:/)=UHl$,&I&)CNb_#%N1,+.+9C

G ′OC+9/e5!%@.!"-0/>,-1@9C,cC%N0N1CC+9/"$'2"#+/w,3"N0C_(E,@DNV[\$'(H!%@./>-0$'2

DVUøCKdf,-0/G[\,B5/.$'@.-0+.C%@GC%2 ("N0B"+NV[\C a/>C%25+.-1$,2

TAcFA

$'BCV

C2`("N0B"+e"C+eC=a/>C2"+.-0$'2"+TTcFT

C/PV

2"$'B"+"$'252"C 5$'2"#B"2"C#%N1,+.+9C"CNb,@9M'C%B"@"C#%N1-0XB"C_2"$'23*$,@.2"!C,U Î%ÈX¾m¿"À5°ÁH ( ;/@CI

O ⊆ TT,FT,PV,TA,FA,CV, IV,DVID]BKAQK+D

TT,FT,PV ⊆ OD]I

GOL/@CIKH13D L]K\+D][aRYa]B :LNLD \m[/N\m[ND FHDNLY?[:]\+^+DNLiF?@EL]I:?1KaNDNL^+][N7F?@CI:?@C[DNLQc3BE/[L

GODNL]IyFHD B :?[*YHD]KX[

FHDja]B @BAQK+Di13/1qO]/[a13NDQcKCy&I&IC$,2^G

Î%ÈX¾m¿"À5°ÁH ( ;/@CIO ⊆ TT,FT,PV,TA,FA,CV, IV,DV

ID]B3AQKDTT,FT,CV ⊆ OD]I

GOL/@CIyKH13D L]K\+D][aRYa]B :LNLD\m[N/N\m[DFHD B :fa]B :LNLDAFHDNL Y?[:]\+^+DNL aN/>\mBE]>D]1+I:?@C[NDNL$FHDNLF?@EL]I:?1KaNDNL

^+][7F?@CI:?@C[DNLQc;BE/[LGO

DNL]IyFHDiB :?[*YHD]KX[iFHD a]B @BAQK+D 13/1 O]/[a13NDQc Î%ÈX¾m¿"À5°ÁH ( ;/@CI

O ⊆ TT,FT,PV,TA,FA,CV, IV,DVID]B#AQKD

TT,FT,TA,FA ⊆O

D]IGO

L/@CIKX13D L]K\+D][aRYa]B :LNLD \m[/N\m[NDFHDNLY?[:]\+^+DNL_LQ:?1`LC5

I:?KX[D7:?K YHD]> D]I1aN/RYHD]> @C1mF?KX@CIcBE/[L

GODNL]IyFHDiB :?[*YHD]KH[ FHDja]B @BAQKD_13/1 O]/[]13NDQc

øC%+e+9C%B"N0C%+#Nb,+9+.C+"CM'@>,(5"C%+G"CKN1,@.M'CB"@á"CK#N1-1XB"C?3H$'@.2"!C,ca"!O"2"-1C+(E,@á"C+eC=a/>C2"+.-0$'2"+G"C+.$,&I&)C/.+y5$'2"2"!C%+%c+.$,2X/ "$'25#I"C+ +9$'B"+ö6Y#%N1,+.+9C%+ "C%+yM'@>(""C%+y"-0+9/w2"#%C+ "!@.!"-0/>,-1@9C%+%cø5CIN0C%B"@9+#%$,&I("N0!%&)C%2:/>+c"C/?"C+M'@>,(5"C%+?+>,25+C5

c"/w,B"@9C,Bºc"M'C%& C/?#%$6gM'C%& -12""B5-0/U

øC/>@>h,-0N("@9!%+9C%2:/>!)(E,@/w,-7/"CNV[\-0"!%C[\BA/>-1N0-1+9C%@KB52"C4"!%#$'&)(*$'+9-0/.-1$'2L(*$'B5@@.!+.$'B"[email protected] C )#°6#%C&IC2:/B"2 ("@.$'35NE %&)C,UHWYNü!=/w,-7/-125-0/>-1,N1C&IC2:/(H$'@/>!4+.B5@_Nb "!#%$'&)(H$'+.-7/>-1$,2^&)$A"B5Nb,[email protected],cZ("B5-1++9B"@"-+º[email protected]%2:/.C%+?M'!%2"!@>,N0-1+./>-0$'2"+C=/?C%25OE2+.B5@?N14"!%#$'&I(H$'+9-0/>-0$'2`C2L#N1-1XB"C+%U;$'B5+`h'$'25+`&I$,2X/.@.!#%$'&)&)C%2:/ N1Ä5!%#%$,&I(H$'+.-7/>-0$'2ý&)$a"B"[email protected] (*CB5/`+9C%@9ha-0@ (H$'B"@)@.!%+9$'B""@9C#%C@9/>,-12"+("@.$'35NE %&)C%+ />C%N0+XB"C 132 "$45>+-)>4 2 c - 1 #0+86 39,)> !)üc 2 "$#&9> 1 @ +

98+-)>: 9> "$45.45"J+8:GU7U7U;$'B5+2"$'B"+_+9$'&)&IC+K+.C@9ha-ü"C4#C/9/>C4"!#%$'&)(*$,+.-0/.-1$'2L(*$,B"@$'35/.C%2"-0@K"C+,N0M'$'@.-7/>"&)C%+IN1-02"!,[email protected]+`(*$,B"@N1C%+("@9$'3"N %&)C%+ !"$#&%'*)'+3#0%,c68%'4 :.øc 6 2 % 2 9>+ )>4 2 C=/@ + 9,)>4;)>4 2 68%,4 :.+9B"@ N1C+M'@>(""C%+ +.,2"+

P5

C/ M,C%& -02""B"-7/U;$'B"+ h,$'2"+ C%2"+9B"-0/.C`df,-0/N1C />@>h,-0N+9B"@_Nb`"!#%$'&)(H$'+.-7/>-1$,2^C%2 #$'B"(HC%+%cø5$'& -12"C aB5-ü2[\(",+_+.$'BAh'C%2:/y!/.!4/>@>-0/>!D89B"+.XB[uo("@9!%+.C2:/UlCNbL^,3H$'B5/.-áC2 (E,@/>-0#%B"N0-1C%@ oB"2 ,N0M'$'@9-0/>5&IC(H$'B"@y#N1#%B5N1C%@yB"2"CI#%$'N0$'@>Ü/>-1$,2Ä&)-125-76&B"&[]B"24M'@.,(""CeC24B5/>-0N1-0+>,2:/+>"!#%$'&)(*$,+.-0/.-1$'2yC%2 #$'B"(HC%+%Ue25OE2 2"$'B"+ümh,$'2"+&)$'2:/>@9!eXB[\B52"C2"$'BAh'C%N0N1CD#Nb,+9+.CD"C("@9$'3"NE &IC+ü"C(E,@/>-7/>-1$,2"2"C%&)C%2:/%c"!=OE2"-0+.+>3"N1C+áC2 N1$'M'-0XB"Ce&)$'2E,"-0XB"Ce"B)+.C 6#%$,2"L$,@."@9C,cE(HC%B5h,C%2:/=/>@.Cy@.!+.$'N0B"+C%2/>C&I("+(H$'N<a2"$,&I-1,N*+9B"@?N1C%+DM,@>,("5C%+?"CNb,@9M'C%B"@D5Cy#N1-0aB5C3H$'@.2"!C,UKB (H$'-02X/)"C haB"C/>"!$'@.-0aB5C,c2"$'B5+ h'$,2"+`h,2"#!LE2"+INV[\!=/>B"5C5CLN1 5!%#%$,&I(H$'+.-7/>-0$'2 3"-7689$'-02X/%cAB52"C2"$'BAh'C%N0N1CM,!%2"!@>,N0-1+>Ü/>-1$,2 "CN1"!#%$'&)(*$,+.-0/.-1$'2)&)$A"B5Nb,[email protected],U<;?$'B"+Dh'$,2"+ehaB#$'&)&IC2X/#%C=/./.Cy5!%#%$,&I(H$'+.-7/>-0$'2 (*CB5/+.C@9ha-1@E2"+N1C#%,+"C%+M'@.,(""C+?+.,2"+M'C& C/?#%$Ü6YM'C& -125"B"-0/%UR25OE2ºc2"$'B5+^h'$,2"+L5!OE2"-y"CBA2"$'B5h,C%N0N1C%+M'!2"!%@.,N1-0+>/.-1$'2"+cN0C%+"!%#$'&I(H$'+9-0/>-0$'2"+

k6Y&)$A"B5Nb,[email protected]^C=/

k63"-0&I$a"B"N1,-1@9C,cXC/?&)$'2:/>@.!_B52"C+.-1&)-1N0-0/.B""Ch,C%#yNb"!#%$'&)(*$,+.-0/.-1$'2 C2L#N1-0aB5C%+%U

Ã2 ("@9!%+9C%2:/>Cá& -12:/>C2E,2:/øXB"C%N0XB"C%+"-1@9C%#=/>-1$,2"+ø"Cå@.C#w"C%@9#w"C,c :,2:/(*$,B"@ø$'3a89C#/>-7d5º[Q,(5(*$'@/>C@"C2"$'BAh'C%N0N1C+?#$'2"2E-1+.+.,2"#C%+?E,2"+N1C_+9BÜ89C/%UWYNG+9C%&35N1C)aB5C`N1C+ M'!2"!%@.,N1-0+>/.-1$'25+y5C Nb"!#%$'&)(*$,+.-0/.-1$'2Ä&)$A"B5Nb,[email protected]+9C`@>,(5("@.$a#w"C%2:/"C N1"!#%$'&)(*$,+.-0/.-1$'2C2Æ#%N0-1XB"C%+U lC%(HC%2"E2X/KB"2"C "-*!%@9C%2"#CydÁ$'25E,&)C%2:/w,N0C"CN1 5!%#%$,&I(H$'+.-7/>-0$'2C%2#%N0-1XB"C+ C+9/XB[]C%N0N1C 2[]C%+/4(E,+yB"2"-0aB5C,c,N1$'@9+aB5C N1C+ 5!%#%$,&I(H$'+.-7/>-0$'2 &I$a"B"N1,-1@9C,cºC2 #%$,B"(*C+C=/3"-Å6089$'-02X/GN1C+.$'2:/%U+lK[]C%+9/?B"+.+9-N1C#,+D(H$'B"@GNb"!#%$'&)(H$'+.-7/>-1$,2 ;KWlKck6Y&)$a"B"[email protected]

k6Y35-1&)$A"B5Nb,[email protected]/#C%N0N1C"C4N1`Nb,@9M'C%B5@K"C4@.,2"M"Uø?2"C4(HC%@.+9(*C#/.-0h'C +9C%@>-0/_"C4#w5C%@.#w5C%@y[QB5/>@9C%+M'!%25!%@>N1-1+./>-0$'2"+"CyNb4"!#%$'&)(*$,+.-0/.-1$'2 &)$a"B"[email protected]"-M:,@.5C%2:/KB52"C #C%@/w,-02"CB"2"-1#-0/.!f#%$,&I&)C_Nb4"!#%$'&)(*$,+.-0/.-1$'2C%2I#%$'B5(*C+GC=/N15!%#%$,&I(H$'+.-7/>-0$'243"-76789$'-12:/ =UXÃ2)(*$'B5@.@>-0/á,B"+9+.-Z!/.B""-0C%@å"C%+G"!%#$'&)(*$'+9-0/.-1$'2"+5$'2:/N1C+åM'@>,(5"C%+å#%$,&I("N /.C%&)C%2:/á"!%#$'&I(H$'+.,3"N1C+2"C?+.$'-0C%2:/á(E,+å"C?Nb,@9M'C%B"@5C?#N1-0aB5C?3H$'@92"!%Cc:#%$'&)&ICNb# 6Y"!#%$'&)(H$'+.-7/>-1$,2UÃ2k`haB«XB"C N0C%+"!%#$'&)(*$'+9-0/.-1$'2"+K2"$'B"+(HC%@.&)C/9/>C2X/º[\$'3A/>C%25-1@"C%+y,N0M'$'@9-0/>5&IC+K"C @.C#%$'2a62E,-0+.+.,2"#%CC/)"C+ N1M'$'@9-0/."&IC+(H$'N<a2"$'&)-bBA (H$'B"@ "C+I(5@.$'3"N %&)C%+ ; ü6g#%$'&)("N0C/>+ C2ýM'!2"!%@.,NôUJ^,-0+(H$'B"@#C%Nb`-0NdfB5/_("@9!,N1,3"N1C&IC2:/K/>@9$'B5h,C%@_"C+_#%Nb+.+.C+_"C4M'@.,(""C+_aB5-ü+.C4("@9/.C%2:/3"-1C2qo

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Décompositions de graphes et algorithmes efficaces

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