On-line vertex-covering

27
Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Décision CNRS UMR 7024 CAHIER DU LAMSADE 183 Septembre 2001 On-line vertex-covering Marc Demange, Vangelis Th. Paschos

Transcript of On-line vertex-covering

Laboratoire d'Analyse et Modélisation de Systèmes p our

l'Aide à la Décision

CNRS UMR 7024

CAHIER DU LAMSADE

183

Septembre 2001

On-line vertex-covering

Marc Demange, Vangelis Th. Paschos

������������ ���������������������� � !"!#�$ �&%('*)�+�% !"!, -&. %�'*/10���+�%(!2/ . ,3 4 )���!5+60�798 . !:%(!5/ . �;) . 0 . /<%()=%(!2/ . � 3> ?;.�@BA ! . 7DCE7F'(%�79G @ +�/�CH7F'I! .�J K !:%(L

t = n>

M�NPO QSRUTWV=X YIZ=Z=[]\(^�_a`bY*TW_a\�Rc[dY*TW_a\e\�f�YIR9gU`;_aR=_a`�YIhjikXl[mTWX&^Fnpo&\(ikXl[]_aR=qUYIhaq�\�[]_rTWV=`sYIq9YI_aR<tmT`�Y*^�_a`�u=`v`�Y*TdodV=_aR=q NwNxNwNwNwNxNwNxNwNwNwNxNwNwNxNwNwNwNxNwNxNwNwNwNxNwNwNxNwNwNwNxNwN M

M�Nzy {�R|\�R}n~ha_aR=X�YIhaq�\�[]_rTWV=`�f2\�[�TWV=Xwo�Y�t]Xt = n

NxNwNwNxNwNwNwNxNwNxNwNwNwNxNwNwNxNwNwNwNxNwN �M�N�M �1\(��Xl[��H\�u=R<�=t�\�R�TWV=X�o&\�`;ZHX&TW_rTW_ri�_rT�g|\�f�YIR9g�YIhaq�\�[]_rTWV=`�f2\�[�TWV=X�o�Y�t]X

t = nNwNwNxNwN �

� ?;.�@BA ! . 7DCE7F'(%�79G�+�/�CH7F'I! .�J K !:%(Ln > t > 2 �� NPO QSR}n~ha_aR=XxikXl[mTWX&^|o&\(ikXl[]_aR=q;��_rTWVn > t > 2

NxNwNxNwNwNwNxNwNwNxNwNwNwNxNwNxNwNwNwNxNwNwN �� Nzy QSR}n~ha_aR=XxikXl[mTWX&^Fnpo&\(ikXl[]_aR=q���_rTWV

t = 2NwNwNwNxNwNxNwNwNwNxNwNwNxNwNwNwNxNwNxNwNwNwNxNwNwN O��

� NzyFNPOGV�Y�t�R=\�_Ptm\�h:Y*TWX��|ikXl[mTW_Po&X�t�NxNwNwNwNxNwNwNxNwNwNwNxNwNxNwNwNwNxNwNwNxNwNwNwNxNwN O��

� NzyFNzy ��hau<tmTWXl[Wt�V�YlikXxR=\;_Pt]\�h:Y*TWX��|ikXl[mTW_Po&X�teNxNwNwNxNwNwNwNxNwNxNwNwNwNxNwNwNxNwNwNwNxNwN O�O� N�M �1\(��Xl[��H\�u=R<�=t�f2\�[�TWV=X�o&\�`;ZHX&TW_rTW_ri�_rT�g|[dY*TW_a\�NxNwNwNxNwNwNwNxNwNxNwNwNwNxNwNwNxNwNwNwNxNwN O(y

� N�M�NPO ��Y�t]Xt > 2

NwNxNwNwNwNxNwNwNxNwNwNwNxNwNxNwNwNwNxNwNwNxNwNwNwNxNwNxNwNwNwNxNwNwN O(y� N�M�Nzy ��_a`�_rTdt�\�R|TWV=Xwo&\�`�ZHX&TW_rTW_ri�_rT�g�f2\�[

t = 2NwNwNwNxNwNwNxNwNwNwNxNwNxNwNwNwNxNwNwN O(y

� # A5A / K ! .�J $ )<+I�E%('*)<+I��! .�J ,F�� # ��!"��� A 7D/ .�@dA ! . 7���/10�7 A $ )<�l7�0 ���¡/ . 7�0 J 7�� , �� ¢ / . + A ���(!2/ . � ,}£# ¤ '*/�/H¥�/E¥�%(L�7F/H'*7�� � 3<¦

� ���}������k�� ������ �����������������������k� ��! �

"$#&%('*)+#,�-/.�02143�576987:<;=6?>A@CBD34-FE�0G6G-CB�H 0�I 6G-CJK8�.�-CLMBONPI @<-RQC573�S<0GTM;UBDBMJK8V57WYXZW[02W[B$BM-\N<69]^WM57BM-Z87B[J�WM8�._QPBMJM`*a

0bI 6GJcJc@CB$NCBdL�:<.�L[@C-CB$NPI B[020GBMJ�3�-eNC34628fQ<57BM-CNF57BdN<BMJ�NCW[LM6GJc6234-CJ�625c57WYXZ3&L[.gS<0GBMJ�Jc@F5�0?.OQh.g5c876GB$>A@C6 XF62B[-Z8NPI iY8c57BV57WMXZWM0GWMBkjb>Z@lI B[020GB=BMJK8�0G.OLM34-Z8c5762S<@F876G34-+N<B=L[BY8c87BVQh.g5c8762BdmO0G.OJc3402@C876234-e>A@CB=0bI 3�-nLM34-CJK8c57@<698Co(p�`qr34@CJsLM34;U;UBM-Ct[34-CJ�Qu._5vWM87@CNC6GBY5v0GB�QC573�S<0GTM;UBrNCBrLM34@FXZBY5c87@C57B�N<BwJc34;U;=BM87JsJc34@CJxNCBM@Fyd;U3&NCTM0GBMJs34-FE0G62-CB�>A@C6<LM3�5c57BMJKQ^34-CN<BM-Z8smzNCBM@CyV;d.�-C62TM57BMJ�N<69]^WM57BM-Z87B[J�NCBx57WMX4W[02BM5�0GBMJ�Jc34;U;=BM87JM`Z{|BvQC57B[;=6GBY5s62;=Q^34JcB>A@CB�02B�1�5�.gQ<:CB�62-CJK8�.�-CL[B�BMJK8w57WYXZWM0GW�Jc3�;U;UBY8rQh.g5rJc3�;U;UBY8[`u}f.g-<J�0GB�JcB[LM34-CN~;=3&NCT[02B�0GB�1�5�._Qh:CB�BMJK857WYXZW[02WrQu._5�Qu.�>A@CBM87JMHCL�:<.4>A@CB�Qh.4>A@CBY8�WY8�.�-Z8�@C-�Jc34@CJKE�1�5�.gQ<:CBf6G-CNC@C628�NCBz0�I 6G-CJK8�.�-CLMBfLM34;=Q<02TM87B�`C��@C6GJMHN<.�-CJv02Br;U3&NCT[02BwQh.g5vQh.4>A@CBY87JMH&-<3�@<Js57BM0?.gy&34-CJv0?.�LM34-Z8c5�.�6G-Z87Br>A@CBr0GBrL�:<3�62ydN<BMJvJc3�;U;UBY87JxNC@$Qh.4>Z@<BY8m�62-<LM0G@F57BrN<.�-CJv0G.�Jc3�0G@F876G34-$B[JK8v695c57WMXZ3AL(.gS<02B4`A��0G@CJsQC57W[LM6GJcWM;UBM-A8v-C34@CJx.�@F873�5762Jc34-CJx@C-n�7�Z�Y�_���7�Z���(���A�r�@C-CB�N<WMLM6GJc6G3�-dJc@F5s@C-UJc34;U;UBY8�QPBM@F8�iM8c57B�;=3&NC62�<W[B�;U3&NC@C0G3f@C-UL[34�F8�>A@C6<NCWYQ^B[-CNUN<@U-C34;�SC57B�1�0G3�Sh.�0NCBdJc3�;U;UBY87J�;U3&NC69�hWMJM`|,�-F�<-e-C34@CJ�WY87@CNC6G34-CJ�@C-e;=3&NCT[02BdJc62;=Q<0GBd34�e02B[Jf6G-CJK8�.�-CLMB[JfJc34-Z8�57WYXZW[02W[BMJ.g57iY87B�Qh.g5r._57iM87B�`

���<�(%Y�7�A��# �=.g0G143�576987:C;UB=�g�h�������^��HA57BML[34@FX&57BM;UBM-Z8sNPI @C-$1�5�._Qh:PHZ5�.gQCQ^3�5c8sNCBwLM34;=Q^WY87628769XF6287W�H45�.gQCQ^3�5c8NPIh.gQCQC573(yF62;d.g876234-P`

� ���l�K�p� � ���l�����P¡v�g�}|���l�h�~�w�

¢D£ %Y�_¤(¥h���¦§-¨3�-CE�026G-CB=LM34;=Q<@F8�.g876234-PH^87:CBV62-<JK8�.g-<LMB�34©�.dQC573�S<0GBM;ª62Jr57BYXZB(.g0GBMN¨JK87BMQCEbSA«&E�JK87BYQn.�-CN¨34-CBV:<.�JMH

.g8w87:CB�B[-CN~34© B(.�L�:~JK87BYQlHu873D625c57BYXZ3&L[.gS<02«kNCB[LM6GNCB�3�-~87:CB�Qh.g5c8�3�© 87:CB��<-<.�0*Jc3402@F876G34-¨NCB[.�0G62-C1$¬�6287:87:C6GJ�JK87BYQ�`O­eBU�<57JK8�JK87@CNF«+87:CB$;U62-<62;�@C;®XZBM5c87BYy&E�LM3(XZBY576G-C1kQC573�S<0GBM;®@C-<NCBY5�8§¬v3¯34-FE�0G62-CBD;=3&NCB[02JLM3�5c57B[JKQ^34-CNC6G-C1D873$8§¬v3ONC69]PBY57BM-A8r¬x.[«FJrXZBY5c876GLMB[J�.g57B�57BYXZB(.g0GBMNl`f°x:<BV©±3�57;UBY5z34-CBV6G;=Q<0G62B[Jr87:<.g8r87:<B6G-FQ<@F8cE�1�5�.gQ<:k6GJ�57BYXZB[.�0GBMN�XZBY5c87BMyAEbS&«AEbXZBM5c87BYy^`s°x:CB�JcBML[3�-<Nk;U3&NCBM0�6G;VQh026GBMJ�87:<.g8�87:CBf62-CQ<@F8cE�1�5�.gQ<:¯62J57BYXZB(.g0GBMN�QPBY5 L[02@<JK87BY57JMHZ6�` B�`GH4Q^BY5�62-<NC@CLMB[N=Jc@FS<1�5�.gQ<:CJ�34©<87:CBv�<-<.�0C1g5�.gQ<:l`*²r-CN<BY5 87:<BvL[02@<JK87BY5cE�;U3&NCBM0bH¬vBd87:CB[-e57BM0?.gy+87:CBDLM34-CJK8c5�.�62-A8�87:<.g8�87:CB$L�:C3462L[B$34©v87:CBdQu._5c8=34©v87:CBd�h-<.�0sJc3�0G@F876G34-RNCB(.�026G-C1¯¬�6287:B[.�L�:=L[02@<JK87BY5�:<.�J 873�S^B�695c57BMXZ3AL(.gS<02B4H4SA«U.�020G3(¬�62-C1zSu.gL�³&8c5�.gL�³F62-C1C`*­RB�.gJcJc@<;=B�87:<.g8�3�-<B�L[.�-UL�:<.�-C14BNCB[LM6GJc6234-CJU@FQ^34-´.~XZBY5c87BMyR;UBM;�S^BM57Jc:C62Q´3�©�87:CBO�<-<.�0�Jc3�0G@F876G34-PH�87:C62JUL�:<.�-C14B�62;=Q<02«F62-<1FHs:C3(¬vBMX4BM5[HJc34;UB�LM34JK8�NCBYQPBM-CNC6G-C1=34-D87:CB�-A@C;�S^BM5�34©l87:CBrXZBY5c876GLMBMJxL�:<.�-C14BMNP`�µ*6G-<.�0G09«$¬vBzJK87@CNF«DJc62;=Q<0GBz;=3&NCB[0¬�:CBM57Bz6G-CJK8�.�-CLMBf62J�57BYXZB[.�0GBMN�B[NC14BYEbSA«&E�B[NC14B�`

¶k·A¸l¹��u¤_º*% �»34-FE�0G6G-CB�.�02143�576987:<;OH<X4BM5c87BYy&E�LM3(XZBY576G-C1CH<LM34;=Q^BY87628769XF628§«$5�.g87623CHh.gQCQC573(yF62;d.g876234-�5�.g87623C`

_a_

� � ���&�k��r�r���F�~��QSR}n~ha_aR=Xwo&\�`;Z=u}TBY*TW_a\�R|_Pt�ikXl[mgbR�Y*TWu=[dYIh�_aR�\�ZHXl[dY*TW_a\�R�YIh1[]X�t]X(YI[WodV6tm_aR<o&XjTWV=Xl[]XxX&^�_PtmT�tm_rTWu�Y*TW_a\�R<t�`;\��}XlhaX��Y�t�Z=[]\��=haXl`�t�f2\�[���V=_PodV�TWV=X��=R�YIh��<Y*TBY*npt]X&T�_Pt�R=\�T�Y�Z=[]_a\�[]_��FR=\(��R�N��pRb\�TWV=Xl[���\�[W�=t�=�<Y*TBY YI[]Xj[]X&ikX(YIhaX��tmTWXlZ}n~�9gFnpt TWXlZ�N�=[]X �9u=XlR9TWhrg��1��V=XlR \�R=X�TW[]_aX�t�TW\|t]\�hrikX�t]u<odV Z=[]\��=haXl`�tS`bYIR9g�T�g�ZHX�tj\�f o&\�R<tmTW[dYI_aR9Tdt��5f2\�[X&^=YI`;Z=haX��F�}X(Y��}ha_aR=X�t \�R TWV=X��=R�YIh�t]\�hau}TW_a\�R��}Xlha_rikXl[mg����}X(Y��}ha_aR=X�t \�R TWV=X�_a`;Z=haXl`�XlR9TBY*TW_a\�R;\�f�TWV=X�t]\�hau}TW_a\�Ro&\�`;Z=u}TWX�����f2\�[Wo&X V=Xl[��*V=_a` TW\bt TBYI[mTjZ=[]\��=haXl`�� tStm\�hau}TW_a\�R �HXlf2\�[]X TWV=X���V=\�haX�tmX&TS\�f¡�<Y*TBY;_PtSo&\�`;Z=haX&TWXlhrg[]X&ikX(YIhaX�� N QSR�TWV=X�\�TWV=Xl[�V�YIR<����TWV=X�t]X�o&\�R<tmTW[dYI_aR9Tdt�`�Ylgw�HX�tmTW[]_Po T�XlR=\�u=q�V�f2\�[Wo&_aR=qxtm\STWV=X�Z=[]\��=haXl` tm\�hrikXl[TW\�_a[][]X&ik\�o�YI�=hrg��}X�o&_P�}Xj\�R�TWV=XSZ�YI[mT�\�f�TWV=X��=R�YIh�tm\�hau}TW_a\�R��}X(YIha_aR=q���_rTWVbX(Y�odV�Z�YI[mT�\�f��<Y*TBY�[]X&ikX(YIhaX����=\�[`�Ylg��HX []Xlh:Y*TW_rikXlhrgD��X(Y�����YIhaha\(��_aR=qbV=Xl[��*V=_a` TW\�q�\b��Y�o���\(ikXl[S�}X�o&_Pt]_a\�R<tSZ=[]X&i�_a\�u<t]hrg6TBY���XlR�YI�H\�u}TSTWV=XZ�YI[mTW_:YIh�tm\�hau}TW_a\�RDo&\�`;Z=u}TWX��6Y*T�X(Y�odV6t TWXlZ�N�1X&T

Π�HX YIR�� ¤ \�Z}TW_a`;_! (Y*TW_a\�R q�[dYIZ=V}n~Z=[]\��=haXl`�N#"�V=X%$'&)(+*-,.&�/�ikXl[Wtm_a\�R \�f

� �}XlR=\�TWX�� �9g6�

Π�H_Pt

TWV=XwZ�YI_a[(Π,R)

��V=Xl[]XR_Pt�Y�t]X&T�\�f�[]u=haX�t��}X(YIha_aR=q;��_rTWV�0

O�N�_aR=f2\�[]`�Y*TW_a\�R�\�R|TWV=Xxi�YIhau=Xw\�f�t]\�`;XxZ�YI[dYI`;X&TWXl[Wt�\�f�TWV=X1�=R�YIh1q�[dYIZ=V��yFN�V=\(�cTWV=X1�=R�YIh�q�[dYIZ=V6_Pt�[]X&ikX(YIhaX�� N

{�R2$�&�(+*3,.&4/jYIhaq�\�[]_rTWV=`65��}X�o&_P�}X�t�Y*T X(Y�odVbt TWXlZ;��V=_PodV�\�fETWV=X��<Y*TBY��2ikXl[mTW_Po&X�t�\�[ X��}q�X�t�� []X&ikX(YIhaX����}u=[]_aR=qTWV=_Pt�tmTWXlZ���_ahah��HXlha\�R=q TW\;TWV=X1�=R�YIh�t]\�hau}TW_a\�R�N7�~Tdt�ZHXl[]f2\�[]`bYIR<o&X�_Pt�`;X(Y�t]u=[]X��|_aR�TWXl[]`�t�\�f�TWV=X t]\�npo�YIhahaX��o&\�`;ZHX&TW_rTW_ri�_rT�g�[dY*TW_a\

cA�}X�=R=X����<f2\�[�YIR|_aR<t TBYIR<o&X

GYIR<��Y�t]X&T

R��Y�t�TWV=Xx[dY*TW_a\ \�f�TWV=Xj��\�[WtmT8�i�YIhau=Xx\�f

TWV=XSt]\�hau}TW_a\�R�o&\�`;Z=u}TWX����9g;_rT���V=XlR�[]u=R=R=_aR=q�\�RGTW\�TWV=X�i�YIhau=XS\�f�Ywt]\�hau}TW_a\�R�o&\�`;Z=u}TWX���\�9Hn~ha_aR=X���_�N X�N!�

�9g YIR�YIhaq�\�[]_rTWV=` []u=R=R=_aR=q�\�R<o&X�TWV=X��=R�YIh�q�[dYIZ=V _Ptjo&\�`;Z=haX&TWXlhrg:�FR=\(��R�N#�pRDTWV=_PtSZ�YIZHXl[S��X��}X(YIh���_rTWV�}X&TWXl[]`;_aR=_PtmTW_Po \�R}n~ha_aR=X YIhaq�\�[]_rTWV=`�tlN�"�V=X�YI�H\(ikX�`�X(Y�tmu=[]X \�f TWV=X��9u�YIha_rT�gD\�f YIRD\�R}n~ha_aR=X YIhaq�\�[]_rTWV=` V�Y�t�HXlXlR�\�[]_aq�_aR�YIhahrg�_aR9TW[]\��}u<o&X���_aR<;rO(y>=?��_aR�\�[W�}Xl[wTW\�tmTWu<��g�Ybf2u=R<�<YI`;XlR9TBYIh o&\�`�Z=u}TWXl[�t]o&_aXlR<o&X�Z=[]\��=haXl`��TWV=X�Z�YIq�_aR=qbZ=[]\��=haXl`�N@�pR9TWXl[]X�tmTW_aR=q�[]X&i�_aX&��tS\�RD\�R}n~ha_aR=X YIhaq�\�[]_rTWV=`�tA�5u=R<�}Xl[jo&\�`;ZHX&TW_rTW_rikX YIR�YIhrg}tm_Pt��SYI[]XZ=[]X�t]XlR9TWX���_aRB;DCE�)F>=~N¡{�hPt]\G�<YIRb_aR9TWXl[]X�tmTWX��H�=R�YIR<o&_:YIh�YIZ=Z=ha_Po�Y*TW_a\�R|\�f�\�R}n~ha_aR=Xxo&\�`;Z=u}TW_aR=q�_Pt�tm\�hrikX���_aRB; �I=~NQSR}n~ha_aR=X�q�[dYIZ=V�Z=[]\��=haXl`�t tmTWu<�}_aX�� u=R9TW_ah R=\(�vYI[]X���TW\6\�u=[J�FR=\(��haX��}q�X���TWV=X;TW[dYlikXlha_aR=q tdYIhaX�tm`�YIRK�L;rOM=N���TWV=Xwq�[dYIZ=V}npo&\�ha\�[]_aR=q��L; �E�1O��E��O>=�YIR<��TWV=Xw_aR<�}XlZHXlR<�}XlR9TjtmX&T��L; � =N�BN�1X&T�u<t;o&\�R<tm_P�}Xl[�Y o&\�`;Z�YIR9g []X�o&Xl_ri�_aR=q `�YIRFu=f"Y�o TWu=[]_aR=q�\�[W�}Xl[Wt f2[]\�` _rTdt;o&ha_aXlR9Tdt&NK"�V=X�t]X�\�[W�}Xl[Wt

V�YlikX�TW\ �HX�Y�olo&XlZ}TWX��\�[�[]XQP X�o TWX�� Y�t�t]\F\�R Y�t�TWV=X&gYI[][]_rikX�N�{�olo&XlZ}TBYIR<o&X�\�[�[]XQP X�o TW_a\�R\�f�TWV=Xb\�[W�}Xl[Wto�YIR �HX�Z=[]\(i�_P�}X�� Xl_rTWV=Xl[x_a`;`;X��}_:Y*TWXlhrg�� _�N X�N!��Y�tjtm\F\�R�Y�txYIR9g6\�[W�}Xl[�YI[][]_rikX�tR�TS'*-UV/MWX&�S�UY,.Z�/\[���� \�[wY*TSTWV=XXlR<����tdYlg \�fjX(Y�odVe`;\�R9TWV]�TS'*^UV/MW�&4S'UY,.Z'/`_baN[�����\�[�YIf:TWXl[bY��}^�X���RFu=`��HXl[�\�fS\�[W�}Xl[Wt2�"tdYlgcO�����\�[W�}Xl[Wt��V�Y�t�YI[][]_rikX��c�TS'*^UV/MW�&4S'UY,.Z'/d_Ea!_e�BNf�pR<o&\�`;Z�Y*TW_a�=_aha_rTW_aX�t��HX&T���XlXlR�\�[W�}Xl[Wtd�"�}u=XxTW\ TWV=Xwf"YI�=[]_Po�Y*TW_a\�R6TW_a`;X��}TWV=X`�Y*TWXl[]_:YIhPt []X �9u=_a[]X���� X&Tdo�N-� YI[]X�Z�YI_a[m��_PtmX�_aR<o&\�`;Z�Y*TW_a�=_aha_rTW_aX�tMg�t]\ Y6t]X&T�\�f�q�ha\���YIhahrg�o&\�`;Z�Y*TW_a�=haX�\�[W�}Xl[Wt_Pt�YIR�_aR<�}XlZHXlR<�}XlR9T t]X&Tw_aR TWV=X \�[W�}Xl[mn~_aR<o&\�`;Z�Y*TW_a�=_aha_rT�g�q�[dYIZ=Vh�2��V=Xl[]X \�[W�}Xl[Wt YI[]X _rTdtSikXl[mTW_Po&X�t��BNR�pf TWV=X\��bP X�o TW_rikX�\�f�TWV=X o&\�`�Z�YIR9g|_Pt�TW\�`bY*^�_a`;_! lXxTWV=X�RFu=`��HXl[�\�f�TWV=X o&\�`;Z�Y*TW_a�=haX \�[W�}Xl[WtxY�olo&XlZ}TWX��D�}u=[]_aR=qG�tdYlg \�R=X�gkX(YI[� t�ZHXl[]_a\��h�5\�[jTWV=X;q�ha\���YIh Z=[]\��}T�_a`�Z=ha_aX�� �9g TWV=Xl`H����TWV=XlR�TWV=X;Z=[]\��=haXl` TW\|�HX�t]\�hrikX���_PtTWV=XB$�&�(+*3,.&4/\iRS j',.i�kEil,.&4m�/onG/X&4m�/M&�U�pM/Ufn�WT$rqM*!/Mi�N`s�\(��X&ikXl[�¡Y�tWtmu=`;X�Y|Z=u=�=ha_Pobo&\�`;Z�YIR9g�\�[ X&ikXlR Yo&\�`;Z�YIR9g \�ZHXl[dY*TW_aR=q ��_rTWV�Z=[]_ri�_ahaXlq�X�� o&ha_aXlR9TdtlN��pR��H\�TWV t]_rTWu�Y*TW_a\�R<tM� TWV=Xbo&\�`�Z�YIR9g�_Pt�o&\�R<tmTW[dYI_aR=X���TW\Y�olo&XlZ}T�YIR9g�\�[W�}Xl[wXl`bYIR�Y*TW_aR=q�f2[]\�` _rTdtwo&ha_aXlR9Tdt�YIR<��V�Y�twXl_rTWV=Xl[wTW\�`�YIRFu=f"Y�o TWu=[]X�\�[W�}Xl[Wtw�9g�_rTdtmXlhaf+��\�[TW\�u<tmX�t]u=�}npo&\�R9TW[dY�o TW\�[WtlN\"�V=XlR���_rTdtx\��bP X�o TW_rikX�_PtxR=\�T�TW\|`�Y*^�_a`�_! lX�TWV=X;Z=[]\��}T ���=u}TwTW\|`�_aR=_a`;_! lX�TWV=Xt]u=�Eo&\�R9TW[dY�o TW_aR=qSo&\kt T��}u=[]_aR=qG��f2\�[�X&^=YI`;Z=haX��*\�R=X¡gkX(YI[� t�ZHXl[]_a\�� N�tF\G�*��_rTWVw[]X�tmZHX�o T�TW\�TWV=X¡_aR<o&\�`�Z�Y*TW_a�=_aha_rT�gq�[dYIZ=V�`;XlR9TW_a\�R=X��\P u<t T�YI�H\(ikX���\�R=X V�Y�tSTW\|`�_aR=_a`;_! lX�TWV=X;o&\�`;Z=haXl`;XlR9Tx\�f�Yb`�Y*^�_a`�u=` _aR<�}XlZHXlR<�}XlR9Tt]X&T �H_�N X�N!��Y�ikXl[mTWX&^Fnpo&\(ikXl[��YIR<��TWV=X Z=[]\��=haXl` TW\�t]\�hrikX _Pt�TWV=X%$'&)(+*-,.&�/Ai�,.&G,.iAkEiuZ'/MWXUV/�j�(Qv�$�Z'/MWX,.&xw�Ny"�V=_Pt_Pt�TWV=XwZ=[]\��=haXl` ��Xw�}X(YIh���_rTWV�_aR�TWV=_Pt�Z�YIZHXl[lN

z _aR=_a`�u=` ikXl[mTWX&^Fnpo&\(ikXl[]_aR=q�Z=[]\��=haXl`��<�}XlR=\�TWX��|�9g|{S� _aR�TWV=XwtmX �9u=Xlh}�<_Pt��}X�=R=X��6Y�t�f2\�haha\(��tM0~w',.Z�/M&SHw'WTS�n��

G(V,E) � v�$'i@n�kbUV/\UY�E/|i�,.&),.i�kEiA(Qv�S�WTm�,.&4S'*-,NUY�2pM/U V ′ ⊆ Vp�k)vM�BUY�ES'U � ∀vivj ∈ E � S'U�*�/�S�p�U$�&4/�$Q��UY�E/

vi, vjq�/M*!$�&Ew'p�UV$

V ′ NK� X6o&\�R<t]_P�}Xl[�TWV�Y*TG

�2��XDt]X&Tn = |V | YIR<�et]u=Z=ZH\ktmX n

�FR=\(��R�o���M�Q�����3���?�>�����L�I�

G � �e�}�o�M�L���3�o�����Q�Q�X�}� �3�>� �?� R �

O

Y*T TWV=X��HXlq�_aR=R=_aR=q \�f�TWV=X�q9YI`;X>�j_Pt []X&ikX(YIhaX���ZHXl[ R=\�R}n~Xl`;Z}T�g�o&hau<tmTWXl[WtM� _�N X�N!� ZHXl[ _aR<�}u<o&X��t]u=�=q�[dYIZ=V<tG1(V1, E1), G2(V2, E2), . . .

\�fG�2��X��}XlR=\�TWX��9g

niTWV=X�\�[W�}Xl[ \�f

Gi�i = 1, 2, . . .

�BN � ikXl[mg�TW_a`;X�YjR=X&�o&hau<tmTWXl[

Gi_Pt�[]X&ikX(YIhaX����<TWV=XwX��}q�X�t�ha_aRG�F_aR=q TWV=XjikXl[mTW_Po&X�t�\�f

Gi��_rTWV�TWV=XxikXl[mTW_Po&X�t�\�f

Gj�j < i

YI[]X YIhPt]\[]X&ikX(YIhaX�� N�� X��}XlR=\�TWX �9g

tTWV=X�RFu=`��HXl[�\�f�o&hau<t TWXl[WtjR=XlX��}X�� t]\�TWV�Y*TSTWV=X ��V=\�haX�q�[dYIZ=V _PtSo&\�`;Z=haX&TWXlhrg

[]X&ikX(YIhaX�� N� XH�=[WtmT;f2\�o&u<t�\�u=[Wt]XlhrikX�t;\�RTWV=X�o�Y�t]X���V=Xl[]X|q�[dYIZ=V _Pt�[]X&ikX(YIhaX�� �9g�`�X(YIR<t�\�f�_rTdt�ikXl[mTW_Po&X�t�YIR<�

o&\�R<t]_P�}Xl[t = n

��_�N X�N!��TWV�Y*TG_Pt;[]X&ikX(YIhaX�� ikXl[mTWX&^Fn~�9gFn�ikXl[mTWX&^EN "�V=_Pt _Pt ��V�Y*T���X�V�YlikXDo�YIhahaX�� S'*^UV/MWX(

&4S'UY,.Z'/%[�_aR \�u=[ o&\�`;Z�YIR9g�`;\��}Xlh��}X�tWo&[]_a�HX�� Z=[]X&i�_a\�u<tmhrgkNH��X�X�tmTBYI�=ha_PtmV Y�q�XlR=Xl[dYIh�[]X�t]u=hrT�YI�H\�u}T TWV=XZHXl[]f2\�[]`�YIR<o&X�\�f�X&ikXl[mg`�_aR=_a`�YIh~{S��n�YIhaq�\�[]_rTWV=` �5_�N X�N!��YIReYIhaq�\�[]_rTWV=` o&\�`;Z=u}TW_aR=q Y `;_aR=_a`�YIh�ikXl[mTWX&^Fno&\(ikXl[�� _aR�o&\�`�Z�YI[]_Ptm\�R���_rTWV;TWV=X�`�Y*^�_a`�u=` `�Y*TdodV=_aR=q�YIhaq�\�[]_rTWV=` f2\�[f{S���k_aR=f2\�[]`�YIhahrg��FTWV=X�[dY*TW_a\�\�f�YIR9g`;_aR=_a`�YIh<ikXl[mTWX&^Fnpo&\(ikXl[�YIq9YI_aR<t T�TWV=X�ikXl[mTWX&^Fnpo&\(ikXl[�o&\�`;Z=u}TWX����9g�TWV=X�`bY*^�_a`�u=` `�Y*TdodV=_aR=q�YIhaq�\�[]_rTWV=` _Pt�H\�u=R<�}X�� YI�H\(ikX �9g

∆/2N��Stm_aR=q�TWV=_Pt�[]X�t]u=hrT �H��X�X�tmTBYI�=ha_PtmVDTWV=X�o&\�`;ZHX&TW_rTW_ri�_rT�g6[dY*TW_a\�\�f¡Y�ikXl[mg6t]_a`;Z=haX

�=u}T�ikXl[mg|R�Y*TWu=[dYIh�\�R}n~ha_aR=X�YIhaq�\�[]_rTWV=` XlR9TWXl[]_aR=q�Y�R=X&��hrg�Z=[]X�tmXlR9TWX���ikXl[mTWX&^v_aR�TWV=X�o&\(ikXl[]_aR=q

Cu=R<�}Xl[

o&\�R<tmTW[]u<o TW_a\�R��1_af¡TWV=Xl[]X X&^�_Pt TdtwYIR X��}q�X�_aR<o&_P�}XlR9TxTW\v�5V=XlR<o&X []X&ikX(YIhaX�� TW\�q�X&TWV=Xl[j��_rTWV

v��TWV=X;YIha[]X(Y���g

[]X&ikX(YIhaX��6XlR<�}ZH\�_aR9Tu\�f���V=_PodVD�}\FX�t�R=\�T��HXlha\�R=q�TW\

C�5f2\�haha\(��_aR=q�\�u=[jY�t]t]u=`;Z}TW_a\�R YI�H\�u}T�TWV=Xw��Ylg

G_Pt�[]X&ikX(YIhaX����uV�Y�t�YI[][]_rikX����HXlf2\�[]X

v�BN

� X&^FT ����X�q�XlR=Xl[dYIha_! lXb\�u=[�tmTWu<��g�Y�tWtmu=`;_aR=qt < n

YIR<��tmTWu<��g�TWV=Xbo&\�`;ZHX&TW_rTW_ri�_rT�g [dY*TW_a\D\�fJ�5`;\�[]Xo&\�`;Z=ha_Po�Y*TWX���� \�R}n~ha_aR=X YIhaq�\�[]_rTWV=`�t�f2\�[���{S�vYIq9YI_aR<tmT|YIR�\�Z}TW_a`�YIh�\�9�n~ha_aR=X YIhaq�\�[]_rTWV=`�N s�Xl[]X|��X6�}_PtmnTW_aR=q�u=_Pt]V T���\�o�Y�t]X�t0

2 < t < nYIR<�

t = 2N �U_rTWV []X�tmZHX�o T;TW\�\�u=[;o&\�`�Z�YIR9g`;\��}Xlh}�¡TWV=X�f2\�[]`;Xl[

[]XlZ=[]X�t]XlR9Tdt#S�*-UV/MWX&�S'UY,.Z'/�_baN[�g�S�*-UV/MWX&�S'UY,.Z'/�_ba!_��FR=\�T�t TWu<�}_aX���V=Xl[]X��}o&\�u=hP��[]XlZ=[]X�t]XlR9T�Ywtm_rTWu�Y*TW_a\�R���V=Xl[]XjYIhahTWV=X�o&hau<tmTWXl[WtjYI[]X \�f�TWV=X�tdYI`;X \�[W�}Xl[lN@"�V=XlR�����X�YIR�YIhrgE lX TWV=X�o�Y�t]X

t = 2Ny"�V=_Pt�o�Y�t]X�V�Y�tM��Y�t���X t]V�YIhah

t]XlX��H_rTdt�\(��RD`�Y*TWV=Xl`�Y*TW_Po�YIh�_aR9TWXl[]X�tmT�Ny<u=[mTWV=Xl[]`;\�[]X��EX&ikXlRD_aR�TWV=X f2[dYI`;X&��\�[L�|\�f�\�u=[So&\�`;Z�YIR9g|`;\��}Xlh_rT�_PtjikXl[mg�R�Y*TWu=[dYIh�N���X&i�_Ptm_rT�TWV=_Ptx`�\��}Xlh YIR<��t]u=Z=ZH\kt]X TWV�Y*T _aR�\�[W�}Xl[xTWV�Y*TwTWV=X;Z=[]\��}u<o Tdt tmTBYI[mTwTW\��HX`�YIRFu=f"Y�o TWu=[]X���� tdYlg�Y*T�TWV=X�_aR<t TBYIR9T

t + 15��\�[W�}Xl[Wt V�YlikX TW\6�HXbYI[][]_rikX��Y*T _aR<t TBYIR9T

tN���u}T f2\�[�t]\�`;X

[]X(Y�t]\�R<t��5X�N q=N!��\�[]q9YIR=_! (Y*TW_a\�R�YIh�\�[xZ=[]\�`;\�TW_a\�R�YIh \�R=X�t�����o&ha_aXlR9TdtwV�YlikX �HXlXlR�q�[dYIR9TWX���t]\�`;X X&^FTW[dY��}Xlh:Ylg��f2\�[wX&^=YI`;Z=haX

t + 10��_aR�\�[W�}Xl[xTW\6tmXlR<��\�[W�}Xl[WtxTW\��HX `bYIRFu=f"Y�o TWu=[]X�� Y*T

t + 15N s�Xl[]X���o&\�`;Z�YIR9g V�Y�t

TW\DYIR<t ��Xl[x_aR�T���\�TW_a`;X�t0#�=[WtmT ��f2\�[jTWV=X;\�[W�}Xl[Wt YI[][]_rikX���u=Z�TW\�_aR<t TBYIR9TtYIR<��t]X�o&\�R<����f2\�[xTWV=X;\�[W�}Xl[Wt

YI[][]_rikX��6f2[]\�`t + 1

TW\t + 10

N~"�V=X�_aR<o&\�`;Z�Y*TW_a�=_aha_rT�g�q�[dYIZ=V<t�\�f�TWV=X�t]XwT���\�t]X&Tdt�\�f�\�[W�}Xl[WtSYI[]XwTWV=X�T���\o&hau<tmTWXl[Wt&N

� X�o&\�R9TW_aRFu=X TWV=X Z�YIZHXl[|�9gUY�t]t]u=`;_aR=q R=\�R}n~_a[][]X&ik\�o�YI�=_aha_rT�g _aR TWV=X�o&\�R<tmTW[]u<o TW_a\�Rc\�fwTWV=X \�R}n~ha_aR=Xt]\�hau}TW_a\�R��k_�N X�N!�9�9g�YIhaha\(��_aR=qx��Y�o��9TW[dY�o��F_aR=q=N�"�V=_Pt�`;X(YIR<t�TWV�Y*T�TWV=X�YIhaq�\�[]_rTWV=` o�YIR _aR9TWXl[WodV�YIR=q�X�Y�RFu=`��HXl[\�f1ikXl[mTW_Po&X�t�_aRbTWV=Xwtm\�hau}TW_a\�R6o&\�`;Z=u}TWX����9g|Y�RFu=`��HXl[�\�f1ikXl[mTW_Po&X�t�R=\�T�_aR<o&hau<�}X��|_aR�_rT�N���u}T���Xxo&\�R<t]_P�}Xl[TWV�Y*T odV�YIR=q�X�t�ZHXl[]f2\�[]`;X�� _a`�Z=hrg�Y�o&\ktmT�\�R�TWV=X�ikXl[mTW_Po&X�t odV�YIR=q�X�� N|"�V=_PtM��_aR�\�u=[�o&\�`;Z�YIR9g X&^=YI`;Z=haX���HX�o&\�`;X�t _aRe�}X�o&_P�}_aR=qG��Y*T;TWV=X�h:Y�tmT�`;\�`;XlR9T � TW\�q�_rikXDtm\�`;X6Y��=�}_rTW_a\�R�YIh�`�YIRFu=f"Y�o TWu=[]_aR=q���\�[L�_aR _rTdtt]u=�}npo&\�R9TW[dY�o TW\�[Wt&N��u}T�tm_aR<o&X�_rT��}\FX�t R=\�T�`;XlX&T�TWV=Xb�}X(Y��}ha_aR=Xbf2\�[�TWV=X�t]X�\�[W�}Xl[WtM� _rT�V�Y�twTW\DZ�Ylg tm\�`;XX&^FTW[dY o&\ktmT�N���XSt TWu<��g�TWV=XSo&\�`;ZHX&TW_rTW_ri�_rT�g���YIq9YI_aR<t T�YIR�\�Z}TW_a`�YIhE\�9�n~ha_aR=XxYIhaq�\�[]_rTWV=`\�¡\�f T���\�YIhaq�\�[]_rTWV=`�tu=R<�}Xl[�T���\;o&\ktmT�`;\��}XlhPtlNf"�V=Xxf2\�[]`�Xl[�_a`;Z=ha_aX�t�TWV�Y*T�TWV=X�o&\ktmT�Z�YI_P�|f2\�[�YIR9g|odV�YIR=q�X�_Pt �}^�X����<��V=_ahaXxTWV=Xh:Y*T]TWXl[�_a`�Z=ha_aX�txTWV�Y*T f2\�[�YIR9g ikXl[mTWX&^�odV�YIR=q�X�� \�R=X�V�Y�twTW\�Z�Ylg�Y|o&\ktmT X �9u�YIh¡TW\�TWV=X;TW\�TBYIh RFu=`��HXl[w\�fikXl[mTW_Po&X�t�odV�YIR=q�X�� N

�_aR�YIhahrg�����X;t TWu<��g�Y|t]ha_aq�V9TWhrg��}_!9EXl[]XlR9T�\�R}n~ha_aR=X;`�\��}Xlh}����V=Xl[]X ��X�Y�tWtmu=`�X�TWV�Y*TwTWV=X;_aR=Z=u}T]n~q�[dYIZ=V_Ptj[]X&ikX(YIhaX���X��}q�X&n~�9gFn~X��}q�X�NR"�\�q�X&TWV=Xl[j��_rTWV TWV=X�YI[][]_ri�YIh \�f�Y�R=X&� X��}q�X���YI[]X []X&ikX(YIhaX�� TWV=X�ha_aRG��tx\�f _rTdtXlR<�}ZH\�_aR9Tdt���_rTWV|TWV=Xx\�R=X�t�\�f�TWV=XwX��}q�X�t�YIha[]X(Y���g|[]X&ikX(YIhaX�� N s�Xl[]X�YIhPtm\ ��Xx�}X&i�_Pt]X�YIR|\�R}n~ha_aR=X�YIhaq�\�[]_rTWV=`YIR<��tmTWu<��g�_rTdt�o&\�`;ZHX&TW_rTW_ri�_rT�g|[dY*TW_a\�YIq9YI_aR<tmTjYIR|\�Z}TW_a`bYIh�\�9Hn~ha_aR=Xx\�R=X�N

� ���&�K� � � ���r�"�C����������r� � 1���E�F�~����"�V=Xwf2\�haha\(��_aR=q��}X�=R=_rTW_a\�R<t���_ahah1�HXwf2[]X �9u=XlR9TWhrg�u<t]X��|TWV=[]\�u=q�V=\�u}T�TWV=XwZ�YIZHXl[lN~<\�[�[]X(Y�tm\�R<t�\�f�haXlq�_a�=_aha_rT�gTWV=X&g�YI[]Xwq�[]\�u=ZHX���V=Xl[]X��<�HXlf2\�[]X�XlR9TWXl[]_aR=q TW\;_rTdt�Z=u=[]XlhrgbTWX�odV=R=_Po�YIh�Z�YI[mT�N� )=%(+IL�! .�J�� { `�Y*TdodV=_aR=q;_Pt�Y tmX&T�\�f�`�u}TWu�YIhahrg|�}_PtYP \�_aR9T�X��}q�X�t�\�f

GN

y

��G1� /E�l7�0 CE7F'(%(!5+k7�� � { ikXl[mTWX&^x_Pt�o�YIhahaX���X&^�ZH\ktmX��w��_rTWVw[]X�t]ZHX�o T�TW\SY�`bY*TdodV=_aR=qM�I_af�_rT�_Pt1R=\�T�XlR<�}ZH\�_aR9T

\�f�YIR9g�X��}q�Xw\�fM�=_aR|\�TWV=Xl[���\�[W�=tM�<_af�_rT�_Pt�R=\�T�tWY*TWu=[dY*TWX��6�9g

MN

# � J ��7 . %*! .�J ��)}%*L � {vZ�Y*TWVvi1 , vi2 , . . . , vik

_Pt�YIu=q�`;XlR9TW_aR=q ��_rTWV[]X�t]ZHX�o T�TW\�Y6`�Y*^�_a`�YIh�`�Y*TdodV}n_aR=q

M_af

k_Pt�X&ikXlR��

vi1 , vik

YI[]XbX&^�ZH\kt]X����_rTWV[]X�t]ZHX�o T�TW\MYIR<��_af

vilvil+1∈ M

�l = 2q

�q = 1, . . . , (k − 2)/2

g�_aR6\�TWV=Xl[���\�[W�=t�<TWV=X�tmX&TM \ {vilvil+1

: l = 2q, q = 1, . . . , (k − 2)/2} ∪{vil+1

vil+2: l = 2q, q = 1, . . . , (k−2)/2} _Pt�YIhPt]\xY�`�Y*TdodV=_aR=q���_rTWV�o�YI[W�}_aR�YIha_rT�g�X �9u�YIhFTW\ |M |+1

N-l. 0�7F� 7 . 0�7 . %��l79% � {�R _aR<�}XlZHXlR<�}XlR9T tmX&T�_Pt�Y�tmu=�<t]X&T�\�f

V ′ ⊆ Vt]u<odV�TWV�Y*T �If2\�[�YIR9g

(vi, vj) ∈ V ′×V ′ �vivj /∈ E

N��! . !"��) A�� '*7��l� ��� � )}G�!5� ) A�� �l79% � { t]X&T ��_ahah<�HXSo�YIhahaX��;`;_aR=_a`�YIh �5[]X�t]Z�N!��`�Y*^�_a`�YIh.����_rTWV�[]X�t]ZHX�o T�TW\

Y�Z=[]\�ZHXl[mT�gπ��_af�_rT�tdY*TW_PtQ�=X�t

��V=_ahaX��}XlhaX&TW_a\�R �5[]X�t]Z�N!��_aR<tmXl[mTW_a\�R��w\�f�YIR XlhaXl`;XlR9T f2[]\�` �5[]X�tmZ�N!�

_aR��S[]X�tmu=hrTdt�_aR6Y�t]X&T�R=\�T�tWY*TW_Pt]f:g�_aR=q

πN

�1)�+�% , � ��r>��� {�R9g �5`�Y*^�_a`�YIh.��_aR<�}XlZHXlR<�}XlR9T�t]X&Tb_Pt;TWV=X o&\�`;Z=haXl`�XlR9T ����_rTWV []X�tmZHX�o TbTW\V��\�f�Y

�5`;_aR=_a`�YIh.� ikXl[mTWX&^Fnpo&\(ikXl[lN�pR ��V�Y*T�f2\�haha\(��t�I��X��}XlR=\�TWX��9g

∆TWV=X�`�Y*^�_a`�u=`��}Xlq�[]XlX�\�f

G�I�9g

τ(G)TWV=X�o�YI[W�}_aR�YIha_rT�g�\�f�Y�`�_aR=_a`�u=`

ikXl[mTWX&^|o&\(ikXl[�\�fG�=�9g

M�5[]X�tmZ�N!�

Mi��Y�`�Y*^�_a`�u=` `bY*TdodV=_aR=q \�f

G�5[]X�t]Z�N!�

Gi��YIR<���9g

P�5[]X�tmZ�N!�

Pi�

TWV=X6t]X&T�\�f�TWV=X�X&^�ZH\kt]X�� ikXl[mTW_Po&X�t�\�fG

�5[]X�tmZ�N!�Gi� ��_rTWV�[]X�t]ZHX�o T�TW\

M�5[]X�t]Z�N!�

Mi�BN� SXlR=\�TWX YIhPt]\

�9gX(M)

�5[]X�t]Z�N!�X(Mi)

� TWV=X�tmX&T�\�f�TWV=XwXlR<�}ZH\�_aR9Tdt�\�fM

�5[]X�t]Z�N!�Mi

�BN�1)�+�% 3 ����r>��� ��\�R<t]_P�}Xl[SY�q�[dYIZ=V

G�G�}^�Y�`bY*^�_a`�YIh `�Y*TdodV=_aR=q

MYIR<�|haX&T

P = V \X(M)N�"�V=XlR�0

O�NP_Pt�_aR<�}XlZHXlR<�}XlR9TSf2\�[

Gg

yFNX(M)

_Pt�Y�ikXl[mTWX&^Fnpo&\(ikXl[�\�fG��_rTWV |X(M)| = 2m

N<\�[

v ∈ V����XD�}XlR=\�TWX �9g

Γ(v)TWV=X tmX&T�\�fxR=Xl_aq�VF�H\�[Wtb\�f

v��_�N X�N!�

Γ(v) = {u : uv ∈ E} � ∆ =max{|Γ(v)| : v ∈ V } N

� � ���l�K�p� � ���l�k���^¡x�g�}����l�<�p�w��� ���F!t = n

� Xf�=[WtmT�o&\�R<tm_P�}Xl[�TWV�Y*TG_Pt�[]X&ikX(YIhaX�� _aR9TW\

t = no&hau<t TWXl[Wt��_�N X�N!�IikXl[mTWX&^Fn~�9gFn�ikXl[mTWX&^EN���Xlf2\�[]X�tmZHX�o&_af:g�_aR=qwYIR

\�R}n~ha_aR=XSYIhaq�\�[]_rTWV=` f2\�[ TWV=_Pt o�Y�t]X��9��X�X�tmTBYI�=ha_Pt]VbYjq�XlR=Xl[dYIh�[]X�t]u=hrT f2\�[�YIR9g;YIhaq�\�[]_rTWV=` �5\�R}n~ha_aR=X�\�[¡\�9�n~ha_aR=X>�o&\�`;Z=u}TW_aR=q�Y�`�_aR=_a`�YIhEikXl[mTWX&^Fnpo&\(ikXl[�H_�N X�N!��Y�ikXl[mTWX&^Fnpo&\(ikXl[�TWV�Y*T�o�YIR=R=\�T��HXw[]X��}u<o&X��|�9g�Xlha_a`;_aR�Y*TW_a\�R�\�ft]\�`;Xx\�f�_rTdt�ikXl[mTW_Po&X�t&N

> �", ?;. %*L�7 )<����'*/FG�!"��)=%(!2/ . '*)}%*!2//H¥�) .�� � ! . !"��) A CH7F'(%(79G @ +k/FCE7F'I! .�J ) A2J /H'I!:%*L�� ) J )�! . �&%��)=G�!5� ��� ��)=%*+*L�! .�J

� X��}XlR=\�TWX��9g��x5������E5���� �"!$#�%jYIR�YIhaq�\�[]_rTWV=` o&\�`;Z=u}TW_aR=qxY�`�Y*^�_a`�u=` `�Y*TdodV=_aR=qM\�f

G�2TWV=X�Z=[]\��=haXl`

\�f��=R<�}_aR=q�Y `�Y*^�_a`�u=` `bY*TdodV=_aR=q�\�f¡Y q�[dYIZ=VD_Pt�ZH\�hrg�R=\�`;_:YIh7�L;rO�OX=N�L�BN ��g|_rTWXl` y�\�f�f"Y�o Twyx�X(M) =

{vi, vj : vivj ∈M} _Pt�Y�ikXl[mTWX&^Fnpo&\(ikXl[�\�f�t]_! lX 2|M | f2\�[ GN& �XlR=\�TWX��9g

mTWV=X�t]_! lXw\�f

MNf�_aR�YIhahrg��HtmX&T

p = |P | = |V \X(M)| N�"�V=Xxf2\�haha\(��_aR=q�haXl`;`�Y���_ahah1�HXwu<tmX��|_aR|TWV=Xl\�[]Xl` O'�'P u<tmT��HXlha\(�1�HYIR<�|h:Y*TWXl[lN' 7�� � ) , � ( $�W S�&)��w�W�S�n��*) ,NUY�E$�kbU7,.p$'*!S'UV/�m\Z'/MWMUY,}v�/Mp � p 6 m(∆ − 1)

a,+.� � ,.& S�mrm�,NUY,}$�& � UY�b/1w'WTSXn��v�$'&)UVS�,.&)pι,.pM$'*!S'UV/�mRZ'/MWMUY,}v�/Mp � UY�E/M& p− ι 6 m(∆− 1)

a¤ '*/�/E¥ � �_r^�YIR�X��}q�X

vivj ∈ Mt]u<odV TWV�Y*T Y*T�haX(Y�tmT�\�R=X�\�f

vi�vjV�Y�t R=Xl_aq�VF�H\�[Wt�_aR

PN6�1X&T

Pi =P ∩Γ(vi) = {pi1 , pi2 , . . . , pi|Pi|

} g Pj_Pt��}X�=R=X��bt]_a`;_ah:YI[]hrgkN tFu=Z=ZH\ktmX |Pi| > |Pj |

N "�V=XlR���_afPj 6= ∅

�FTWV=Xf2\�haha\(��_aR=q;V=\�hP�=tM0

{

Pj ⊂ Pi

|Pj| = 1

�mOI�

M

�pR f"Y�o T �1t]u=Z=ZH\kt]XPj 6⊆ Pi

NJ"�V=XlR�� TWV=Xl[]X X&^�_PtmTdtwY*TxhaX(Y�tmTx\�R=X�ikXl[mTWX&^pjk∈ Pj \ Pi

YIR<��Y*TwhaX(Y�t Tw\�R=XikXl[mTWX&^

pil ∈ Pi \ PjN#"�V=X�Z�Y*TWV

pil , vi, vj , pjk

_PtxYIu=q�`;XlR9TW_aR=qb��_rTWV []X�t]ZHX�o TjTW\M��o&\�R9TW[dY��}_Po TW_aR=q�t]\

TWV=Xwf"Y�o T�TWV�Y*TM_Pt�`�Y*^�_a`�u=`�N

QSR�TWV=X \�TWV=Xl[�V�YIR<���Et]u=Z=ZH\ktmXPj ⊂ Pi

�=u}T |Pj| > 1N�"�V=XlR�����_rTWV�TWV=X�tdYI`;X�YI[]q�u=`;XlR9Tdt�Hf2\�[jYIR9g

T���\�ikXl[mTW_Po&X�tpil

YIR<�pjk

_aRPj ∩ Pi

�}TWV=XwZ�Y*TWVpil , vi, vj , pjk

_Pt�YIu=q�`;XlR9TW_aR=q ��_rTWV|[]X�t]ZHX�o T�TW\MN

��\�R<tm_P�}Xl[�R=\(�cTWV=Xwf2\�haha\(��_aR=q;q�[dYIZ=VB(N,EB)

o&\�R<t TW[]u<o TWX�� Y�t�f2\�haha\(��t0• f2\�[wX&ikXl[mg�X��}q�X;\�f M

��X��}[dYl� YbikXl[mTWX&^<�5haX&TNM

�HX;TWV=X�tmX&T \�f�ikXl[mTW_Po&X�t tm\6�}[dYl��R���g���X�YIhPt]\o&\�R<t]_P�}Xl[�ikXl[mTW_Po&X�t�\�f

PY�t�ikXl[mTW_Po&X�t�\�f

Ng=_aR�\�TWV=Xl[���\�[W�=t

N = NM ∪ Pg

• haX&T nij�HX6TWV=X6ikXl[mTWX&^�\�f

NMY�t]t]\�o&_:Y*TWX��e��_rTWV

vivj ∈ Mg�_afjTWV=Xl[]XDX&^�_PtmTdt�YIR X��}q�XDha_aRG�F_aR=q

Xl_rTWV=Xl[vi\�[

vj��_rTWVDY ikXl[mTWX&^

pk ∈ P�}TWV=XlR

nijpk ∈ EBN

��g�X&^�Z=[]X�tWtm_a\�R �mOI��YIR<�bTWV=Xw�}_PtWo&u<t]t]_a\�RDYI�H\(ikX��HYIR9gnij ∈ NM

_Pt�ha_aRG��X�����_rTWV6Y*T�`;\kt T∆− 1

ikXl[mTW_Po&X�t\�f

P�2TWV=Xx`�Y*^�_a`�u=`v�}Xlq�[]XlXw\�f

G_Pt

∆YIR<��\�R=XwX��}q�X � TWV=Xw`�Y*TdodV=_aR=q \�R=X � ha_aRG��t vi

��_rTWVvj�BN�"�V=_Pt

[]Xl`�YI_aR<t�TW[]u=X�_afPj = ∅ N���\�R<t]X �9u=XlR9TWhrg�� p 6 m(∆− 1)

NQSR�TWV=X�\�TWV=Xl[�V�YIR<����_af

Go&\�R9TBYI_aR<tSY t]X&T

I\�f

ι_Ptm\�h:Y*TWX���ikXl[mTW_Po&X�tM��TWV=XlR|TWV=X�YI[]q�u=`;XlR9TS�}X&ikXlha\�ZHX��

YI�H\(ikXw[]Xl`�YI_aR<t�i�YIha_P�|\�R�TWV=X�q�[dYIZ=VG′(V \ I, E)

����N X�N � NtF_aR<o&X

n = 2m + pYIR<���=�9gbhaXl`;`bY�O'�

p 6 m(∆− 1)�5[]X�t]Z�N!�

p− ι 6 m(∆− 1)���}\�R=XwX(Y�t]_ahrgbq�X&Tdt

n 6 m(∆ + 1)�5[]X�t]Z�N!�

n− ι 6 m(∆ + 1)��YIR<�|[]X(Y�odV=X�t�TWV=Xxf2\�haha\(��_aR=q�haXl`;`�Y�N

' 7�� � ) 3 � +�& S�&)� w�W�S�n�� ) ,NUY�%&�$|,.pM$'*!S'UV/�m�Z'/MWXUY,}v�/Xp � m > n/(∆ + 1)a +.�1UY�E/#w�W�S�n��H�bS�p

ι,.pM$�*�S'UV/�m

Z'/MWXUY,}v�/Xp � UY�E/M& m > (n− ι)/(∆ + 1)a

� L�7F/H'*7�� , � ( $�WAS�&G� w'WTS�n��G � UY�E/JWTS�UY,}$H$Q�1UY�E/Jp�,�� / $Q�dS�&)��i�,.&),.iRS'*�Z'/MWMUV/Tj�(Qv�$�Z'/XWJUV$|UY�E/dp�,��I/A$Q�UY�E/JZ'/MWXUV/�j�(Qv�$�Z'/MW#,.&4m�k)v�/�mHqX������������ �����������������,.pdq�$�kE&4m�/�m\SxqT$�Z'/ qX�

∆/2a

¤ '*/�/E¥ � {St]t]u=`;XJ�=[WtmTSTWV�Y*TGV�Y�t�R=\�_Pt]\�h:Y*TWX��DikXl[mTW_Po&X�tM���}XlR=\�TWX��9g

CY;`;_aR=_a`�YIh�ikXl[mTWX&^6o&\(ikXl[j\�f

GYIR<�D�9gsTWV=X�tm_! lX�\�f�TWV=X _aR<�}XlZHXlR<�}XlR9T�tmX&TxY�tWt]\�o&_:Y*TWX��6��_rTWV

C�H_�N X�N!� |V \ C| = s

N SXlR=\�TWX �9gmTWV=X

t]_! lX�\�f�Y�`�Y*^�_a`�u=` `�Y*TdodV=_aR=qM\�f

GN ��g6_rTWXl` y�\�f¡f"Y�o T yx��YIhaq�\�[]_rTWV=` �x5 �����E5���� �"!$#�%�_aR<�}u<o&X�twY

ikXl[mTWX&^Fnpo&\(ikXl[�\�f�t]_! lX2mN���X�o�YIhahe�=R�YIhahrg�TWV�Y*T �<�9g�_rTWXl` Oj\�f�f"Y�o Tjyx�

P = V \X(M)_Pt�_aR<�}XlZHXlR<�}XlR9T�N

tF_aR<o&XC_Pt�t]u=Z=ZH\ktmX�� `;_aR=_a`bYIh}�1TWV=X�_aR<�}XlZHXlR<�}XlR9T t]X&T

V \ C_Ptw`�Y*^�_a`�YIh�N|��\�R<tmX �9u=XlR9TWhrg�� u<tm_aR=q

s > n/(∆ + 1)�L; M�=N���}��Xxq�X&T

|C| = n− s 6 (∆ + 1)s− s = ∆s�~y��

� XbR=\(� �}_Pt TW_aR=q�u=_Pt]V TWV=X�f2\�haha\(��_aR=q T���\ o�Y�tmX�t �}XlZHXlR<�}_aR=q�\�RTWV=X�i�YIhau=X�t�\�fsYIR<�

m0:�5_.�

s 6 m�

�5_a_.�s > m

N<\�[�o�Y�tmX

s 6 m�=u<t]_aR=q;X&^�Z=[]X�tWtm_a\�R �~y����=��Xxq�X&T 0

|C|2m

6∆s

2m6

2

��M��<\�[�o�Y�t]X

s > m�=u<t]_aR=q;haXl`;`�Y�O'�}��Xxq�X&T 0

|C|2m

=n− s

2m6

2m + p−m

2m6

m + m(∆− 1)

2=

2

� � ���\�`��=_aR=_aR=q X&^�Z=[]X�tWt]_a\�R<t���M���YIR<� � � ����\�R=XxZ=[]\(ikX�t�TWV=XjTWV=Xl\�[]Xl`�N��\�R<tm_P�}Xl[�R=\(� TWV�Y*T

Go&\�R9TBYI_aR<t Y�tmX&T

I\�f�_Pt]\�h:Y*TWX���ikXl[mTW_Po&X�t&NR"�V=XlR��

C�HXl_aR=q|`;_aR=_a`bYIh}��_rT��}\FX�t

R=\�T o&\�R9TBYI_aR�YIR9g _Pt]\�h:Y*TWX���ikXl[mTWX&^ENR<u=[mTWV=Xl[]`;\�[]X��MAX � MATCHING(G[V \ I]) = MAX � MATCHING(G)

Ns�XlR<o&X�TWV=XSYIR�YIhrg}t]_Pt�ZHXl[]f2\�[]`�X��1P u<t T�YI�H\(ikX�[]Xl`�YI_aR<t¡i�YIha_P�bYIR<��o&\�R<o&hau<�}X�t�TWV=X�Z=[]\F\�f�\�f TWV=X�TWV=Xl\�[]Xl`�N

�~Tj_Pt���Xlhahrn?�FR=\(��R �L; M�=N��TWV�Y*T � f2\�[jYIR9g�q�[dYIZ=VGYIR<�Df2\�[jYIR9g�`bY*^�_a`�YIh�`�Y*TdodV=_aR=q

M�5\�f o�YI[W�}_aR�YIhrn

_rT�gm��\�f

G�

τ(G) > m�~���

��\�R<tmX �9u=XlR9TWhrg��H�9g�TWV=Xl\�[]Xl` O'�=TWV=Xwf2\�haha\(��_aR=q�o&\�[]\�hah:YI[mg�V=\�hP�=t�_a`;`�X��}_:Y*TWXlhrgkN

¢ /�'*/ A5A )<' � , � ( $'W#S�&G�Jw�W�S�n�� � UY�E/�WTS�UY,}$ $Q�@UY�b/�p�,��I/1$Q��S'&G��iA,.&G,.iRS�*�Z'/XWMUV/�j�(Qv�$�Z'/MW�UV$�UY�b/�p�,��I/1$Q�yUY�b/$�n�UY,.i�S'* $�&4/J,.pdqT$'kE&�mr/�m\Srq�$�Z'/AqX�∆a

> �:3 # . / .�@BA ! . 7 ) A5J /�'I!2%(L�� ¥�/H';%*L�7 +�)��l7t = n

� X�YI[]Xw[]X(Y���g�R=\(�cTW\;t]ZHX�o&_af:g6YIR<�6YIR�YIhrgE lXwTWV=Xxf2\�haha\(��_aR=q;\�R}n~ha_aR=X�YIhaq�\�[]_rTWV=`�f2\�[���{S�SN��� %�!$#����������

C← ∅ �t← 1 �Gt ← ({vt}, ∅) �� �"! � �

t < n � �v← vt+1 �t← t + 1 �Et ← Et ∪ ��������������������������� ��� �!�"�$#$���#%�'&��(��)�������� Vt �Vt ← Vt ∪ {vt+1} �Gt ← (Vt, Et) �! * ∃u 6= v : uv ∈ Et ∧ u /∈ C

��� � #C← C ∪ {v} *�! �� ��,+��"-'+�� � �� # �/. ���������

¤ '*/H� /E��!:%*!2/ .c, �10 �E/�v�$�i@nG/UY,NUY,.ZI,NUY�JWTS'UY,}$A$Q�@S'*Dwr$�W�,NUY�ri32 465 �dS wrS�,.&)p�U S�&`$�n�UY,.i�S'*4$87 (+*-,.&4/�S'*Dwr$�W�,NUY�ri�M$�W:96;c,.p�qT$'kE&�mr/�mASrqT$'Z�/1qX�

∆ � ) ��,N*!/@,NUYp@v�$�i�nb/UY,NUY,.ZI,NUY�dWTS'UY,}$ S wrS�,.&)p�U � �������� ������� �=<+S�p�p�kEi�,.&Ew�UY�ES'U���������� ����������#W�/MUYkbW�&)p�qT$'UY�|SJiRSIj�,.iAkEi6iRS'UVvM��,.&EwM

S�&4mX(M) > ,.p�qT$'kE&�mr/�mASxqT$�Z'/1qX� ∆/2

a@?y$'UY�qT$'kE&�m'pJS�WT/dUY,!w��xUVa

¤ '*/�/E¥ � <\�haha\(��_aR=q YIhaq�\�[]_rTWV=`A������G� ∀v ∈ C� ∃uv ∈ E

t]u<odV�TWV�Y*Tu /∈ C

N�s�XlR<o&X���TWV=X�ikXl[mTWX&^Fnpo&\(ikXl[Co&\�`;Z=u}TWX���_Pt�`;_aR=_a`�YIh�N "�V=XlR���YIZ=Z=ha_Po�Y*TW_a\�R<t�\�f�o&\�[]\�hah:YI[mg�OxYIR<�b\�f�TWV=Xl\�[]Xl` O'�}[]X�t]ZHX�o TW_rikXlhrg��Ho&\�R<o&hau<�}X

TWV=Xw[dY*TW_a\kt�o&h:YI_a`;X�� N�_r^�R=\(� Y

∆ ∈ N�=o&\�R<tm_P�}Xl[�Y�t TBYI[

S∆+1\�R

∆ + 1ikXl[mTW_Po&X�tlN�Q��9i�_a\�u<t]hrg��

τ(S∆+1) = 1N tFu=Z=ZH\ktmX

TWV�Y*T�_rTdt o&XlR9TWXl[ _Pt TWV=X��=[WtmT¡ikXl[mTWX&^;[]X&ikX(YIhaX���gFTWV=X�[]X�t T�\�fHikXl[mTW_Po&X�t�o�YIR��HX�[]X&ikX(YIhaX���_aR�YIR9g \�[W�}Xl[lN "�V=XlR��YIhaq�\�[]_rTWV=`B���������_ahah9R=\�T�_aR<o&hau<�}X TWV=X�t TBYI[mnpo&XlR9TWXl[�_aR

C�I��V=_ahaX _rT���_ahah9_aR<o&hau<�}X�YIhah9TWV=X¡[]Xl`bYI_aR=_aR=q�ikXl[mTW_Po&X�t

\�fS∆+1

N|"�V=Xl[]Xlf2\�[]X���TWV=X�o&\�`�ZHX&TW_rTW_ri�_rT�g�[dY*TW_a\DY�odV=_aX&ikX�� _aR�TWV=_Pt�o�Y�t]X�_PtwX �9u�YIh TW\∆N|<u=[mTWV=Xl[]`;\�[]X��

t]_aR<o&Xτ(S∆+1) = m = 1

�}TWV=X�o&\�`;ZHX&TW_rTW_ri�_rT�g�[dY*TW_a\∆/2

YIq9YI_aR<t T �x5 ��� �x5���� ��! #�%;_Pt�YIhPt]\�Y�odV=_aX&ikX�� N

> �:> ' / K 7F' $ /H� . 0�� / . %*L�7 +�/H� � 79%*!:%*!:C�!:% � /E¥�) . � ) A2J /H'I!:%(L�� ¥~/�'�%*L�7D+�)��l7t = n

tFu=Z=ZH\kt]X�TWV�Y*T ikXl[mTW_Po&X�t�YI[]X�RFu=`��HXl[]X���_aR;TWV=X�\�[W�}Xl[ TWV=X&g�YI[][]_rikX�g�_aRbt TWXlZi�kikXl[mTWX&^

vi_Pt []X&ikX(YIhaX�� N¡{�hPtm\

o&\�R<t]_P�}Xl[�TWV�Y*T ��_aR�tmTWXlZi� {v1, . . . , vi} = Ci ∪ Si

����V=Xl[]XCi�}[dYl��twTWV=X;ikXl[mTWX&^Fnpt]X&T�_aR<o&hau<�}X���_aR�TWV=X

ikXl[mTWX&^ o&\(ikXl[ u=R<�}Xl[ o&\�R<tmTW[]u<o TW_a\�RbYIR<�Si = {v1, . . . , vi}\Ci

N�"�V=X7�=R�YIh<q�[dYIZ=V _Pt¡�}XlR=\�TWX�����Y�t u<t]u�YIhahrg���9g

G(V,E)YIR<� _rTdtj`bY*^�_a`�u=` �}Xlq�[]XlX;�9g

∆NA"�V=X Z=u=[]ZH\ktmX;\�f TWV=_Ptwt]X�o TW_a\�R�_PtjTW\|Z=[]\(i�_P�}X ha_a`�_rTdtj_aR

TWV=X;o&\�`;ZHX&TW_rTW_ri�_rT�g ��YIq9YI_aR<tmT�YIR�\�Z}TW_a`�YIh \�9Hn~ha_aR=X�YIhaq�\�[]_rTWV=`\�S\�f�YIR9g \�R}n~ha_aR=X�YIhaq�\�[]_rTWV=` tm\�hri�_aR=q6��{S���_rTWV

t = n�5\(ikXl[jYIhah�TWV=X���Ylg}t�TWV=X _aR=Z=u}T]n~q�[dYIZ=V _Pt�[]X&ikX(YIhaX����BN��1X&T�u<t�o&\�R<t]_P�}Xl[�TWV=X t]\�hau}TW_a\�RD\�f¡��{S�

Y�t�Y�T���\�n~Z=h:YlgkXl[Wt q9YI`�X�����V=Xl[]X�TWV=X|�=[WtmT�\�R=X��5Z=h:YlgkXl[�OI�x[]X&ikX(YIhPt�TWV=X�_aR<t TBYIR<o&X�YIR<��TWV=XbtmX�o&\�R<��\�R=X�5Z=h:YlgkXl[jy���o&\�R<tmTW[]u<o Tdt�TWV=X�tm\�hau}TW_a\�R�N "�V=XlR��=��XxZ=[]\(ikXwTWV=Xxf2\�haha\(��_aR=q;TWV=Xl\�[]Xl`�N� L�7F/H'*7�� 3 �

[�aDC1$%S�* wr$�WX,NUY��i v�S�&cS�vM��,}/MZ'/Rv�$�i@nG/UY,NUY,.Z ,NUY�HWTS'UY,}$\p�UYWX,}vMU}*-�`qT/U}UV/MWdUY�ES�&∆ � /MZ'/X&B, ��S�&B,.p$�iR$�W+n���,}v$Q�

G,.pFEI&�$�) &2,.& S�m�Z'S�&4v�/Ia

_EaDC1$HS'* wr$�WX,NUY��i v�S�& S�vM��,}/MZ'/Av�$'i@nG/MUY,NUY,.ZI,NUY�|WTS�UY,}$Rp�UYWX,}vU}*3�%qT/U}UV/MWJUY�ES�&∆− 2 � /MZ'/M&:, � G

,.pdS�UYW�/�/S�&4m

n,.pDEI&4$�) &2,.& S�m�Z'S�&4v�/Ia

¤ '*/�/E¥ /E¥�!:%(7�� , � "�V=X�_Pt]\�`;\�[]Z=V=_Po|\�fG[]X&ikX(YIhaX�� _aReY���i�YIR<o&X6o&\�R<t]_Pt Tdt;\�fSY �}_PtNP \�_aR9Tbo&\�hahaX�o TW_a\�R

\�fptmTBYI[WtM��X(Y�odV6\�f�\�[W�}Xl[

∆ + 1YIR<��\�f

∆− 1_Ptm\�h:Y*TWX���ikXl[mTW_Po&X�tM����V=Xl[]X

∆YIR<�

pYI[]X1�}^�X���_aR9TWXlq�Xl[Wt&N

QS�9i�_a\�u<tmhrg��FTWV=X��}Xlq�[]XlX�\�fG_Pt

∆YIR<�;_rTdt¡\�[W�}Xl[

n = p(∆+1)+∆−1N�{St]t]u=`;X�TWV�Y*T Z=h:YlgkXl[SO�[]X&ikX(YIhPt

TWV=Xwq�[dYIZ=V|��_rTWV|[]X�t]ZHX�o T�TW\ TWV=Xwf2\�haha\(��_aR=q;[]u=haX�t0! _af

Cio&\�R9TBYI_aR<t

∆_Pt]\�h:Y*TWX���ikXl[mTW_Po&X�t��5f2\�[ TWV=X�q�[dYIZ=V|YIha[]X(Y���g�[]X&ikX(YIhaX�������TWV=XlR

vi+1_Pt�ha_aRG��X��;TW\�YIhah

TWV=X�t]XxikXl[mTW_Po&X�tMg!5! _af

vi ∈ Si�5_aR�\�TWV=Xl[ ��\�[W�=t�

viV�Y�t�R=\�T��HXlXlR�TBY���XlRb_aR�TWV=XStm\�hau}TW_a\�R���YIR<�

vi_Pt R=\�T�ha_aRG��X���TW\�YIR9g

ikXl[mTWX&^vj�j < i

�HYIR<��_afi 6 n−∆

�}TWV=XlR|ikXl[mTW_Po&X�tvi, vi+1, . . . , vi+∆

f2\�[]` Y tmTBYI[�[]\F\�TWX��|_aRvig

!5!5! _afptmTBYI[Wt�V�YlikXw�HXlXlR�[]X&ikX(YIhaX����<TWV=Xw[]X�tmT�\�f�TWV=XjikXl[mTW_Po&X�t�[]X&ikX(YIhaX��DYI[]Xw_Ptm\�h:Y*TWX���g

!:C _af�[]u=haX�t ! YIR<� !5! o�YIR=R=\�T��HX�YIZ=Z=ha_aX�� YIR<�i 6 n− 1

�=TWV=XlR�ikXl[mTWX&^vi+1

_Pt�_Ptm\�h:Y*TWX�����_rTWV6[]X�tmZHX�o TTW\ TWV=Xxq�[dYIZ=VDYIha[]X(Y���g|[]X&ikX(YIhaX�� N

{�Z=Z=ha_Po�Y*TW_a\�R<t�\�f�[]u=haX�t ! TW\ !2C YI�H\(ikX�_a`;Z=ha_aX�tjTWV�Y*T�Z=h:YlgkXl[�y�o�YIR=R=\�T��}\|�HX&T]TWXl[wTWV�YIR o&\(ikXl[]_aR=q6X��}q�X�t\�f YIR9g�tmTBYI[��9g�_rTdt�haX(YlikX�tM����V=_ahaX�\�Z}TW_a`�YIh�\�9Hn~ha_aR=X�t]\�hau}TW_a\�RDo&\�R<t]_PtmTdt�\�f�TWV=X tmTBYI[mnpo&XlR9TWXl[Wt&N�"�V=Xl[]Xlf2\�[]X�Y[dY*TW_a\;\�f

∆_Pt�Y�odV=_aX&ikX��DY*T��HX�t TSYIR<��TWV=_Pt�o&\�`;Z=haX&TWX�t�TWV=XwZ=[]\F\�f�\�f�_rTWXl` Oj\�f�TWV=XxTWV=Xl\�[]Xl`�N

¤ '*/�/E¥�/E¥S!:%(7�� 3 � �1X&T∆�HX�YIRD_aR9TWXlq�Xl[Sq�[]X(Y*TWXl[�TWV�YIR�� \�[�X �9u�YIh�TW\G� M�YIR<�Dt]X&T

n = ∆(∆ + 1) + 1N

��\�R<tm_P�}Xl[�TWV�Y*T�Z=h:YlgkXl[�Oj[]X&ikX(YIhPt�TWV=Xxq�[dYIZ=V6f2\�haha\(��_aR=q;TWV=Xw[]u=haX�t��HXlha\(�J0� ! � _af

Cio&\�R9TBYI_aR<t

∆_Ptm\�h:Y*TWX���ikXl[mTW_Po&X�ty�2��_rTWV []X�tmZHX�o T TW\STWV=X�q�[dYIZ=V;YIha[]X(Y���g�[]X&ikX(YIhaX���� YIR<�

i 6 n−2�

TWV=XlRvi+1

_Pt�ha_aRG��X��|TW\�YIhah�TWV=X�tmX�_Ptm\�h:Y*TWX��|ikXl[mTW_Po&X�tg� !5! � _af

vi ∈ Si�5_aR�\�TWV=Xl[ ��\�[W�=t�

viV�Y�t�R=\�T��HXlXlR�TBY���XlRb_aR�TWV=XStm\�hau}TW_a\�R���YIR<�

vi_Pt R=\�T�ha_aRG��X���TW\�YIR9g

ikXl[mTWX&^vj�j < i

��YIR<�D_afi 6 n − ∆ − 1

� TWV=XlR ikXl[mTW_Po&X�tvi, vi+1, . . . , vi+∆

f2\�[]` YbtmTBYI[j[]\F\�TWX��_aR

vig

� !5!5! � o&\�R<t]_P�}Xl[vi ∈ Si

�vi_Pt]\�h:Y*TWX��D��_rTWV []X�tmZHX�o TxTW\�TWV=X q�[dYIZ=V�YIha[]X(Y���g []X&ikX(YIhaX�����YIR<�

n − 2 > i >

n − ∆g t]X&T

A = {vj : j < i, vj ∈ Ci,∀k 6 i, vjvk /∈ E} �5_�N X�N!� A_PtxTWV=X�tmX&T \�f�TWV=X�_Ptm\�h:Y*TWX��

ikXl[mTW_Po&X�t�HY*T�_aR<tmTBYIR9Tt�=TBY���XlR�_aR

Ci��YIR<�

B = {vi+2, . . . , vn−1}g}TWV=XlR�0

��Y��vi+1

_Pt�ha_aRG��X���TW\viYIR<��TW\�YIR9g�XlhaXl`�XlR9T�\�f�tmX&T

Ag

�5����TWV=XwXlhaXl`;XlR9Tdt�\�fBf2\�[]` YIR�_aR<�}XlZHXlR<�}XlR9TjtmX&TSYIR<�6YI[]Xwha_aRG��X���TW\

vi

� !:C � _af []u=haX�t � ! � YIR<� � !5! � �}\|R=\�T�YIZ=Z=hrg�YIR<� _afi 6 n − ∆

��TWV=XlR ikXl[mTWX&^vi+1

_Ptj_Pt]\�h:Y*TWX�� ��_rTWV TWV=Xq�[dYIZ=VDYIha[]X(Y���g|[]X&ikX(YIhaX���g

� C �vn_Ptxha_aRG��X���TW\

∆ikXl[mTW_Po&X�t�\�f��}Xlq�[]XlX O�Z=_Po���X���_aR�TWV=X�tmX&ikXl[dYIh�o&\�R=R=X�o TWX�� o&\�`;ZH\�R=XlR9Tdt�\�f�TWV=X

q�[dYIZ=V�[]X&ikX(YIhaX���u=R9TW_ah�tmTWXlZn− 1

N�pR �=q�u=[]X O�Y q�[dYIZ=V

G�}T]TW_aR=q�[]u=haX�t � ! � TW\ � C � _Pt;t]V=\(��R�f2\�[

n = 31YIR<�

∆ = 5N {¡Xl[mTW_Po&X�t�YI[]X

RFu=`��HXl[]X��6_aR�TWV=X�\�[W�}Xl[�TWV=X&g�V�YlikX �HXlXlRD[]X&ikX(YIhaX�� Nys�Xl[]X��H[]u=haX � !5!5! � _Pt�YIZ=Z=ha_aX��Df2\�[�ikXl[mTWX&^ y��FN�"�V=XlR��A = {19, 20} YIR<� B = {29, 30} N�"�V=Xxo&_a[Wo&haXSikXl[mTW_Po&X�t�[]XlZ=[]X�tmXlR9T�TWV=Xx��{S��npt]\�hau}TW_a\�R��}��V=_ahaXSTWV=XxtL�9u�YI[]X\�R=X�t�[]XlZ=[]X�tmXlR9T TWV=Xb_aR<�}XlZHXlR<�}XlR9T�t]X&T Y�tWtm\�o&_:Y*TWX����_rTWV_rT�N { q�[dYIZ=V �}T]TW_aR=q _rTWXl` O�\�f�TWV=XbTWV=Xl\�[]Xl`o&\�u=hP�b�HXxY�t�TWV=XS\�R=XS\�fe�=q�u=[]X O�_aR<�}u<o&X��b�9g;TWV=XxtmX&T�\�f1ikXl[mTW_Po&X�t {1, . . . , 27} Z=hau<t�\�R=XS_Pt]\�h:Y*TWX���tT�9u�YI[]XikXl[mTWX&^EN�pf�[]u=haX � !"!5! � _Pt�R=\�T YIZ=Z=ha_aX����kTWV=XlR _aR tmTWXlZ

n−1��TWV=X�q�[dYIZ=V;o&\�R9TBYI_aR<t

∆tmTBYI[WtM�kTWV=Xl_a[�ikXl[mTW_Po&X�t \�fH�}Xlq�[]XlXwO

`�Y��F_aR=q|Z�YI[mTw\�f TWV=X;t]\�hau}TW_a\�R o&\�R<tmTW[]u<o TWX����9g Z=h:YlgkXl[�yFNA�pR TWV=_Ptwo�Y�t]X��τ(G) = ∆ + 1

�2TWV=X []\F\�Tdtx\�f

F

31

����

����

����

����������������������������

����������������

����������������

����������������

������������������������

����������������������������

����

����

����

����

����������������������������

������������������������

����������������

!!!!!!!!!!!!!!!!

""""""""""""""""

########################

$$$$$$$$$$$$$$$$$$$$$$$$%%&&

''((

))**

++,,

--..

//00111111111111111111111111

222222222222222222222222

3333333333333333

4444444444444444

5555555555555555

6666666666666666

777777777777777777777777

8888888888888888888888889999::::;;<<

==>>

??@@

AABB

CCDD

EEFF

GGHH

IIJJ

KKLL

MMNN

OOPP

QQRR

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT

UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU

VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV

WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

]]^^

____________________________________________________________________________________________________________________________________

````````````````````````````````````````````````````````````````````````````````````````````````````````````````````````````````````

aaaaaaaaaaa

bbbbbbbbbbb

ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd

eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

ffffffffffffffffffffffffffffffffffffffffffffffffffffff

gggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggg

hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh iiiiiiiiiii

jjjjjjjjjjj

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk

llllllllllllllllllllllllllllllllllllllllllll

mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn

ooooooooooooooooooooooooooooooooooooooooooooooooooooooo

ppppppppppppppppppppppppppppppppppppppppppppppppppppppp

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

1

2 3 4 5 6

7 8 9

18

10

11 12 13 14

15

16

17 19

20

21

22 23 24 25 26

27

28 29 30

sstt

�_aq�u=[]X�O'0 { q�[dYIZ=V��}T]TW_aR=q�[]u=haX�t � ! � TW\ � C � ��_rTWVn = 31

YIR<�∆ = 5

N

TWV=X t TBYI[Wt�Z=hau<t�ikXl[mTWX&^vn���<��V=_ahaXwTWV=X t]\�hau}TW_a\�R o&\�R<t TW[]u<o TWX��D_Pt�\�f t]_! lX

∆(∆ + 1)N~"�V=X�o&\�`;ZHX&TW_rTW_ri�_rT�g

[dY*TW_a\;_Pt�_aR|TWV=_Pt�o�Y�t]X∆N

tFu=Z=ZH\kt]X�R=\(� TWV�Y*Tj[]u=haX � !5!"! � _PtSYIZ=Z=ha_aX�� �5[]X�o�YIhah�TWV�Y*T�TWV=_Pt�_Pt�TWV=X�o�Y�t]X f2\�[�ikXl[mTWX&^ y�� _aR`�=q�u=[]X�OI�BN"�V=XlR;_aR�tmTWXlZ

i�9TWV=X�q�[dYIZ=V�o&\�R<t]_Pt Tdt�\�f

ktmTBYI[Wt@�2TWV=Xl_a[�haX(YlikX�t `�Y��F_aR=q�Z�YI[mT�\�fETWV=X�tm\�hau}TW_a\�Rbo&\�R<tmTW[]u<o TWX��

�9g�Z=h:YlgkXl[�y�� Z=hau<t;TWV=XDikXl[mTW_Po&X�t�\�fA ∪ {i} N �pR�TWV=_Pt�o�Y�tmX���TWV=X6TW\�TBYIh�RFu=`��HXl[�\�fjikXl[mTW_Po&X�t�ikXl[]_!�=X�t

n = ∆(∆ + 1) + 1 = k(∆ + 1) + |A| + |B| + 3�H��_rTWV |A| < ∆

�5_af¡R=\�T � []u=haX � ! � ��\�u=hP�D�HX YIZ=Z=ha_aX��\�R=X�`;\�[]X;TW_a`;X>�wYIR<� |B| 6 ∆ − 2

�5�HX�o�YIu<tmXi > n − ∆

�BNH��Xb�}X��}u<o&Xk = ∆ − 1

N��pR TWV=_Pt o�Y�tmX��τ(G) = ∆ − 1 + 3 = ∆ + 2

����V=_ahaX�TWV=X t]\�hau}TW_a\�Rh�=R�YIhahrg o&\�R<t TW[]u<o TWX�� �9g�Z=h:YlgkXl[|y V�Y�tbY*T�haX(Y�tmT(∆− 1)∆ + ∆ = ∆2 ikXl[mTW_Po&X�t1�2TWV=XjhaX(YlikX�t�\�f�TWV=X ∆− 1

t TBYI[Wt�Z=hau<t�t]X&TAZ=hau<t�t]X&T

BZ=hau<t

xi+1�BN�"�V=X

o&\�`;ZHX&TW_rTW_ri�_rT�g�[dY*TW_a\;_a`;Z=ha_aX���_Pt�TWV=XlRDY*T�haX(Y�tmT∆− 2

N��pR%�=q�u=[]X�O'�τ(G) = 7

�2TWV=Xx\�Z}TW_a`bYIh1tm\�hau}TW_a\�R�_PtTWV=X t]X&T {1, 10, 18, 21, 27, 28, 31} �=��V=_ahaXwTWV=X�tm\�hau}TW_a\�R o&\�R<tmTW[]u<o TWX��6�9g�Z=h:YlgkXl[jy��2TWV=X t]X&T�\�f�TWV=X o&_a[Wo&haXikXl[mTW_Po&X�t�\�f �=q�u=[]X�OI��_Pt�\�f�o�YI[W�}_aR�YIha_rT�gDy��FN

�pRb\�[W�}Xl[�TW\�o&\�R<o&hau<�}X��=haX&T�u<t�R=\�TWXSTWV�Y*T�TWV=Xjq�[dYIZ=V|Y*T�t TWXlZn− 1

o&\�R<t]_PtmTdt�\�f∆Y�o g}o&ha_Pojo&\�R=R=X�o TWX��

o&\�`;ZH\�R=XlR9Tdt��X(Y�odV�\�fETWV=Xl` o&\�R9TBYI_aR=_aR=q Y*T�haX(Y�tmT�\�R=X�ikXl[mTWX&^;\�f1�}Xlq�[]XlX�O�N "�V=_Pt `�Y���X�t TWV�Y*T�[]u=haX � C � _Ptf2X(Y�t]_a�=haXjYIR<� q�u�YI[dYIR9TWXlX�t�TWV�Y*T¡TWV=Xy�=R�YIhHq�[dYIZ=V�_Pt�YjTW[]XlX�NftF\G�9TWV=X�Z=[]\F\�f5t \�f _rTWXl` yxYIR<�;\�fETWV=X�TWV=Xl\�[]Xl`YI[]X�o&\�`;Z=haX&TWX�N

u � ���l�K�p� � ���l�k���^¡ �}����l�<�p�w��� ���F!n > t > 2

� X Y�t]t]u=`;X�_aR TWV=_Pt�tmX�o TW_a\�RcTWV�Y*TG_Pt|[]X&ikX(YIhaX��c�9g R=\�R}n~Xl`;Z}T�g o&hau<tmTWXl[Wt

Gi�

i = 1, . . . , t����_rTWV

2 6 t < nN���X��=[WtmT�t TWu<��g�TWV=Xxo�Y�t]X

t > 2N�"�V=Xjo�Y�t]X

t = 2�}�HXl_aR=q _aR9TWXl[]X�tmTW_aR=q �9g�_rTdt]Xlhaf+�=_Pt�X&^=YI`;_aR=X��

t]XlZ�YI[dY*TWXlhrg|_aR6tmX�o TW_a\�R � NzyFN ��Xwt]u=Z=ZH\ktmXwTWV�Y*Tt_Pt �FR=\(��R6Y*T�TWV=Xw�HXlq�_aR=R=_aR=q�\�f�TWV=Xwq9YI`;X�N

� �", ?;.�@BA ! . 7DCE7F'(%�79G�+�/�CE7F'I! .�J K !:%*Ln > t > 2

<\�[�TWV=X�o�Y�t]Xx��Xw�}X(YIh1��_rTWV�_aR�TWV=_Pt�tmX�o TW_a\�R��<��XxZ=[]\�ZH\ktmXwTWV=Xwf2\�haha\(��_aR=q�YIhaq�\�[]_rTWV=`�N��� %�!$#�� � �"��������wv &�&�� # v �$��� G1

�xzyw{

C← X(MAX�MATCHING(G1)) ��wv &�&�� # v �$��� Gi, i = 2 . . . , t

x � { *%���i← 2

��t � �x�� {

C← C ∪ X(MAX � MATCHING(Gi)) �x�� { *���u ∈ Vi \ X(MAX

�MATCHING(Gi)) � �x� { !!* ∃v ∈ (∪16j6i−1Vj) \ C : uv ∈ E

��� � #C← C ∪ {u} *"!

x { � �x�� { � �x�� { �,+��"-'+�� �� # � � � �"�������

QS�9i�_a\�u<tmhrg���TWV=XDt]X&TC�=R�YIhahrgeo&\�`;Z=u}TWX����9g � ���'�����_PtbY ikXl[mTWX&^Fnpo&\(ikXl[�SYIhrTWV=\�u=q�V R=\�TbR=X�o&X�t]tdYI[]_ahrg`;_aR=_a`�YIh�N�tF\G�1Z=[]\�ZH\kt]_rTW_a\�R O �}\FX�twR=\�Tw[]XlZ=[]X�t]XlR9T�TWV=X���\�[WtmT�o�Y�t]X;f2\�[x_rTdtwo&\�`;ZHX&TW_rTW_ri�_rT�g [dY*TW_a\=N � \�TWX

TWV�Y*T|f2\�[bTWV=X�o�Y�t]X ��V=Xl[]X o&hau<tmTWXl[Wt6YI[]X�Y�tWt]u=`;X�� ��_rTWV=\�u}T6YIR9g []X�tmTW[]_Po TW_a\�R��jt]X&T]TW_aR=q |C| 6 n − |I|�2��V=Xl[]XI�}XlR=\�TWX�t;TWV=X�t]X&T�\�fS_Pt]\�h:Y*TWX�� ikXl[mTW_Po&X�tM��_afjYIR9g)�;YIR<� u<t]_aR=q�haXl`;`bY�yx��o&\�`;ZHX&TW_rTW_ri�_rT�g [dY*TW_a\

∆ + 1_Pt�_a`;`;X��}_:Y*TWXlhrg��}X��}u<o&X�� N

� L�7F/H'*7�� >�� /MUλi

q�/2UY�E/:&Gkbi�qT/MW�$L��UY�b/:,.p$'*�S�UV/�m Z'/MWXUY,}v�/Xp2$Q�Gi

,.&�UYWT$Im�k)v�/�m ,.&C

S'&�m p/MUλ =

∑ti=1 λi

S'&�mρ = λ/|A| a 0 �E/M& � UY�E/Bv�$�i@nG/UY,NUY,.ZI,NUY� WTS'UY,}$ $Q�2S'*Dw�$'W�,NUY��i � �2,4 5 � S wrS�,.&)p�URS�&

$�n�UY,.i�S'* 96;KS'*Dw�$'W�,NUY��i ,.p�qT$�kE&4m�/�m\SrqT$'Z�/ q��2 + (∆− 2)/(2− ρ)

a¤ '*/�/E¥ � SXlR=\�TWX6�9g

Ai�

i = 2, . . . , t��TWV=X|ikXl[mTWX&^Fnpt]X&Tdt�_aR9TW[]\��}u<o&X��e_aR

C�}u=[]_aR=q�TWV=X6X&^�X�o&u}TW_a\�Re\�f

ha_aR=X�t x�� { TW\ x { \�f�YIhaq�\�[]_rTWV=` � �"������;YIR<�|t]X&T A = ∪ti=2Ai

N�QS�<tmXl[mikXjTWV�Y*T�ikXl[mTWX&^Fnpt]X&TA_Pt�X&^�ZH\kt]X��

��_rTWV�[]X�t]ZHX�o T TW\STWV=X1�5R=\�R}n~`bY*^�_a`�u=`\��`�Y*TdodV=_aR=q ∪ti=1Mi

g9`;\�[]Xl\(ikXl[�k_rT¡�}\FX�t R=\�T¡o&\�R9TBYI_aR�YIR9g _Pt]\�h:Y*TWX��ikXl[mTWX&^EN�QS�<tmXl[mikXjYIhPt]\�TWV�Y*T�YIR9g;_Ptm\�h:Y*TWX�ikXl[mTWX&^;_Pt X&^�ZH\ktmX��;��_rTWV�[]X�t]ZHX�o T�TW\�YIR9g;`bY*TdodV=_aR=q�\�f

GgFV=XlR<o&X

λ 6 |A| ⇐⇒ ρ 6 1�}F��

SXlR=\�TWX��9gMi

�i = 1, . . . , t

� Y;`�Y*^�_a`�u=`�`bY*TdodV=_aR=q�\�fGi�Et]X&T

mi = |Mi|�i = 2, . . . , t

N@"�V=XlR��HTWV=Xo�YI[W�}_aR�YIha_rT�g|\�f�TWV=Xw\�R}n~ha_aR=X�t]\�hau}TW_a\�R

Co&\�`;Z=u}TWX��|�9g � ���'�����o�YIR|�HXx��[]_rT]TWXlRDY�t�f2\�haha\(��tM0

|C| = 2t∑

i=1

mi + |A| �~���

�1X&TE′ �HX�TWV=X�tmX&TS\�f X��}q�X�t�TWV�Y*TSV�YlikX XlR9TBYI_ahaX��D_aR9TW[]\��}u<o TW_a\�R \�f�TWV=X�ikXl[mTW_Po&X�tS\�f A _aR C

YIR<�D�}XlR=\�TWX�9g

B(A ∪ (V1 \X(M1)), E′)TWV=XxZ�YI[mTW_:YIh�tmu=�=q�[dYIZ=V6\�f

G_aR<�}u<o&X��|�9g

A ∪ (V1 \X(M1))YIR<���9g

E′ N{�hPtm\G� �}XlR=\�TWX|�9gMB

YD`�Y*^�_a`�u=` `�Y*TdodV=_aR=q \�fBYIR<��9g

mBTWV=X�o�YI[W�}_aR�YIha_rT�g\�f

MBN ��Xl`�YI[L�

YIhPt]\|TWV�Y*TMB ∪t

i=1 Mi_Pt�Y�`�Y*TdodV=_aR=q�\�f

G��R=\�T�R=X�tWt]X�o�YI[]_ahrg�`�Y*^�_a`�u=``��YIR<� TWV�Y*T�TWV=X q�[dYIZ=V

B_Pt

�=_aZ�YI[mTW_rTWX���_rTWVbo&\�ha\�[mnpo&h:Y�t]t]X�tAYIR<�

V1 \X(M1)N � ^�Z=[]X�tWtm_a\�RB�~��� o�YIR���f2\�[¡TWV=X�Z=u=[]ZH\ktmX�t�\�f�\�u=[ Z=[]\F\�f+�

�HXw[]X&��[]_rT]TWXlR Y�t

τ(G) >

(

t∑

i=1

mi

)

+ mB�}O��

YIR<���<u<t]_aR=q;X&^�Z=[]X�tWtm_a\�R<t��~����YIR<�c�}O����=TWV=X�o&\�`;ZHX&TW_rTW_ri�_rT�g|[dY*TW_a\;\�f � ���'����;�HX�o&\�`;X�t

ct � OLVC =|C|

τ(G)6

|C|(

t∑

i=1mi

)

+ mB

=

2t∑

i=1mi + |A|

(

t∑

i=1mi

)

+ mB

= 2 +|A| − 2mB

(

t∑

i=1mi

)

+ mB

�}C��

�_aR�YIhahrg��<u<t]_aR=q;haXl`�`�Y yx�}��Xxq�X&T

mB >|A ∪ (V1 \X(M1))|

∆>|A|∆

�mO����

O

QSR�TWV=X�\�TWV=Xl[¡V�YIR<���9haX&Ti ∈ {1, . . . , t} YIR<� haX&T I

�HX�TWV=X�t]X&T¡\�f�TWV=X�_Pt]\�h:Y*TWX���ikXl[mTW_Po&X�t¡\�fEo&hau<tmTWXl[GiN ��X

TWV=XlR;V�YlikX |I| = λi+(|I|−|Ai∩I|) �k\�[ |Ai|−λi = |Ai|+(|I|−|Ai∩I|)−|I| 6 pi−|I| 6 mi(∆−1)�

��V=Xl[]X6TWV=X:�=[Wt T�_aR=X �9u�YIha_rT�geV=\�hP�=t��HX�o�YIu<t]XAi ⊆ Pi

YIR<����Y�t;��X6V�YlikX YIha[]X(Y���g `;XlR9TW_a\�R=X��e_aReTWV=X�HXlq�_aR=R=_aR=q�\�fSTWV=X6Z=[]\F\�f+��TWV=X6t]X&T

I \ (Ai ∩ I)���HXl_aR=q _Pt]\�h:Y*TWX��e_aR

Gi��_rT�_Pt;X&^�ZH\ktmX�����_rTWVe[]X�tmZHX�o T

TW\DYIR9g `bY*TdodV=_aR=q�\�fGig�TWV=X�t]X�o&\�R<��_aR=X �9u�YIha_rT�g�V=\�hP�=txTWV�YIRG��twTW\�haXl`;`�Y O�NHtFu=`;`;_aR=q|_aR=X �9u�YIha_rTW_aX�t

|Ai| − λi 6 mi(∆− 1)f2\�[

i = 1, . . . , t�}��Xx\��}TBYI_aR

t∑

i=1

mi >|A| − λ

∆− 1

�mO�OI�

�Stm_aR=q;X&^�Z=[]X�t]t]_a\�R<t��}F���� �mO�����YIR<�c�mO�OI���<X&^�Z=[]X�tWtm_a\�R �}C����HX�o&\�`�X�t

ct � OLVC 6 2 +|A| − 2mB

(

t∑

i=1mi

)

+ mB

6 2 +|A| − 2|A|

∆|A|−λ∆−1 + |A|

6 2 +∆−2∆

∆(1−ρ)+∆−1∆(∆−1)

= 2 +(∆− 2)(∆− 1)

∆(2− ρ)− 16 2 +

∆− 2

2− ρ

�mO(y��

YIR<��TWV=XwZ=[]\F\�f�\�f�TWV=XjTWV=Xl\�[]Xl`�_Pt�o&\�`;Z=haX&TWX�N��X�o�YIhah�TWV�Y*T

ρ 6 1�5X&^�Z=[]X�t]t]_a\�R �}F��L�BNf<u=[mTWV=Xl[]`;\�[]X��<X&^�Z=[]X�tWtm_a\�R �mO(y���_Pt�_aR<o&[]X(Y�t]_aR=q ��_rTWV

ρg=V=XlR<o&X��

t]X&T]TW_aR=qρ = 1

TWV=X�f2\�haha\(��_aR=q;[]X�t]u=hrT�_Pt�_a`�`;X��}_:Y*TWXlhrgb\��}TBYI_aR=X�� N¢ /�'*/ A5A )<' � 3 � 0 �E/Av�$�i@nG/UY,NUY,.Z ,NUY�|WTS'UY,}$\$L�JS'*Dwr$�WX,NUY��i � �2,4 5 ��S wrS�,.&)p�U S�& $�n�UY,.iRS'* 9 ;hS'*Dwr$�W�,NUY�ri ,.pqT$'kE&�mr/�m\Srq�$�Z'/ q��

∆a

QSR6TWV=X \�TWV=Xl[SV�YIR<����o&\�R<tm_P�}Xl[STWV�Y*Txo&hau<t TWXl[WtxYI[][]_rikX ��_rTWV=\�u}Tj_Pt]\�h:Y*TWX��6ikXl[mTW_Po&X�t&N#�pR6TWV=_PtSo�Y�t]X��Ef2\�[xYIR9gi = 1, . . . , t

�λi = 0

�<t]\G�ρ = 0

YIR<���<u<t]_aR=q;X&^�Z=[]X�tWtm_a\�RO(yx�}TWV=Xwf2\�haha\(��_aR=q;V=\�hP�=tlN¢ /�'*/ A5A )<' � > � � �E/M&4/XZ'/MW�v*3kEpXUV/MW�p`S�WXW�,.Z'/ ) ,NUY�E$�kbUd,.pM$'*!S'UV/�m Z'/MWMUY,}v�/Mp � UY�E/�v�$�i�nb/UY,NUY,.ZI,NUY� WTS'UY,}$ $Q�S'*Dwr$�W�,NUY�ri ���2,4 5 ��S wrS�,.&)p�U S�& $�n�UY,.iRS�* 96;KS'* wr$�WX,NUY��i ,.pdq�$�kE&4m�/�mHSrqT$�Z'/AqX�

(∆ + 2)/2a

� \�TWX�TWV�Y*T TWV=X6tm\�hau}TW_a\�RCo&\�`;Z=u}TWX�� �9g YIhaq�\�[]_rTWV=` � ���'�����_Pt�R=\�T;R=X�o&X�tWtdYI[]_ahrg `;_aR=_a`�YIh�Ne��\�R<t]X&n�9u=XlR9TWhrg��<TWV=Xx[]X�t]u=hrT�\�f�o&\�[]\�hah:YI[mgDy�o�YIR=R=\�T��HX��}Xl[]_rikX��|�9g|�}_a[]X�o TjYIZ=Z=ha_Po�Y*TW_a\�R6\�f�o&\�[]\�hah:YI[mg�O�N

"�V=X��H\�u=R<� \�f�o&\�[]\�hah:YI[mg�M�o�YIR��HX�t]ha_aq�V9TWhrg�_a`�Z=[]\(ikX����9gxTWV=X�f2\�haha\(��_aR=qS��YlgkN tF_aR<o&X���f2\�[i = 1, . . . , t

�TWV=XxikXl[mTW_Po&X�t�\�f

AiYI[]X�X&^�ZH\ktmX��|��_rTWV|[]X�tmZHX�o T�TW\

Mi�=u<t]_aR=q�haXl`;`�YbO��

Ai ∩ I = ∅ ���}��Xxq�X&T|Ai| 6 mi(∆− 1)

�mO�M��QSRDTWV=X�\�TWV=Xl[jV�YIR<����tm_aR<o&X R=\�o&hau<tmTWXl[wo&\�R9TBYI_aR<tx_Ptm\�h:Y*TWX�� ikXl[mTW_Po&X�t� TWV=X �=_aZ�YI[mTW_rTWX q�[dYIZ=V

B(A ∪ (V1 \X(M1)), E

′)o&\�R<t]_P�}Xl[]X��b_aR�TWV=X�Z=[]\F\�f�\�f TWV=Xl\�[]Xl` MwV�Y�t `�Y*^�_a`�u=` �}Xlq�[]XlXj�H\�u=R<�}X���YI�H\(ikX��9g

∆− 1��Y*T�haX(Y�t T�\�R=X X��}q�X;ZHXl[xikXl[mTWX&^ _aRA ∪ (V1 \ X(M1))

�1ha_aRG��tx_rTxTW\�ikXl[mTW_Po&X�t�\�fX(Mi)

�i = 1, . . . , t

�BN��\�R<tmX �9u=XlR9TWhrg��

|A| 6 (∆− 1) |V1 \X(M1)| 6 (∆− 1)mB�mO � �

��V=Xl[]X�TWV=X6h:Y�tmT�_aR=X �9u�YIha_rT�g V=\�hP�=t��HX�o�YIu<t]X���_aRUY��=_aZ�YI[mTW_rTWXDq�[dYIZ=V���TWV=XDo�YI[W�}_aR�YIha_rT�ge\�fwY�`�Y*^�_a`�u=``�Y*TdodV=_aR=q�_Pt�tm`bYIhahaXl[�TWV�YIR�TWV=X o�YI[W�}_aR�YIha_rT�g6\�f�TWV=X tm`�YIhahaX�tmTjYI`;\�R=q TWV=X o&\�ha\�[mnpo&h:Y�tWtmX�tlNytF\G�Eo&\�`��=_aR=_aR=qX&^�Z=[]X�tWtm_a\�R<t��mO�M���YIR<�c�mO � ���<X&^�Z=[]X�tWtm_a\�R �}C����HX�o&\�`;X�t0

ct � OLVC 6 2 +|A| − 2mB

(

t∑

i=1mi

)

+ mB

6 2 +|A| − 2 |A|

∆−12|A|∆−1

= 2 +∆− 3

2=

∆ + 1

2

�mO(���

YIR<�|X&^�Z=[]X�tWtm_a\�R �mO(����haX(Y��=t�_a`;`�X��}_:Y*TWXlhrg�TW\ TWV=Xxf2\�haha\(��_aR=qR�=R�YIh�o&\�[]\�hah:YI[mgkN¢ /�'*/ A5A )<' ��� � � �E/M&4/XZ'/MW�v*3kEpXUV/MW�p`S�WXW�,.Z'/ ) ,NUY�E$�kbUd,.pM$'*!S'UV/�m Z'/MWMUY,}v�/Mp � UY�E/�v�$�i�nb/UY,NUY,.ZI,NUY� WTS'UY,}$ $Q�S'*Dwr$�W�,NUY�ri ���2,4 5 ��S wrS�,.&)p�U S�& $�n�UY,.iRS�* 96;KS'* wr$�WX,NUY��i ,.pdq�$�kE&4m�/�mHSrqT$�Z'/AqX�

(∆ + 1)/2a

C

� �:3 ?;.�@BA ! . 7DCE7F'(%�79G @ +�/�CH7F'I! .�J K !:%*Lt = 2

tFu=Z=ZH\kt]XbR=\(� TWV�Y*T�TWV=Xb_aR=Z=u}T�q�[dYIZ=V_Pt []X&ikX(YIhaX�� ��_rTWV=_aR T���\ o&hau<tmTWXl[WtG1(V1, E1)

YIR<�G2(V2, E2)

N{St]t]u=`;X�YIhPtm\|TWV�Y*T

n�1TWV=X�\�[W�}Xl[�\�f�TWV=XR�=R�YIh q�[dYIZ=V���_Pt1�FR=\(��R�Y*TwTWV=X;�HXlq�_aR=R=_aR=qD\�f�TWV=X;q9YI`;X�NR��X

[]X�o�YIhah�TWV�Y*T �1f2\�haha\(��_aR=q�\�u=[wY�tWtmu=`�Z}TW_a\�R<tM�1\�R=X V�Y�tSTW\|�}X�o&_P�}X���V=_PodV ikXl[mTW_Po&X�tx\�f¡TWV=X �=[Wt T�o&hau<t TWXl[x��_ahah�HXlha\�R=q;TW\�TWV=X1�=R�YIh�t]\�hau}TW_a\�R6�HXlf2\�[]XxTWV=X YI[][]_ri�YIh1\�f�TWV=X�t]X�o&\�R<��o&hau<t TWXl[lN

� �:3 �5,GL�)�� . /�!5�l/ A )}%(7�0�CE7F'(%(!"+�7��

�pR TWV=_Pt�t]X�o TW_a\�R���X�t]u=Z=ZH\ktmX�TWV�Y*T�R=\xY��=�}_rTW_a\�R�YIh=V9g�ZH\�TWV=X�tmX�t¡YI[]X�Y��}`;_rT]TWX�� \�R TWV=X�f2\�[]`�t�\�f=TWV=X�o&hau<tmTWXl[WtYIR<�6YIR�YIhrgE lXxTWV=X o&\�`;ZHX&TW_rTW_ri�_rT�g|[dY*TW_a\;\�f�TWV=Xwf2\�haha\(��_aR=q�YIhaq�\�[]_rTWV=`�N��� %�!$#�� � �"��������wv &�&�� # v �$��� G1

�xzyw{ ��5 � �

V1 6 n/2 � � C← V1 �x � { %"� �� x { �x�� {V1 > n/2 � � C← X(MAX

�MATCHING(G1)) �x�� { %"� �� x�� { �x� { � ��wv &�&�� # v �$��� G2

�x { �,+��"-'+��

C← C ∪ X(MAX � MATCHING(G2)) �x�� { �,+��"-'+��C← C ∪ V2 �� # �/. � � �"�������

� L�7F/H'*7�� � � +.�G

�ES�pR&�$:,.p$'*�S�UV/�m`Z'/MWMUY,}v�/Mp � UY�E/M& UY�E/%v�$�i�nb/UY,NUY,.ZI,NUY�2WTS'UY,}$�$L�|S�* wr$�WX,NUY��i���(2 465 �S wrS�,.&)p�U7S�&c$�n�UY,.iRS'* 96;hS�* wr$�WX,NUY��i Z'/XWX, � /Xp

c2 � OLVC 6 (∆ + 5)/2a 0 ��,.p�WTS'UY,}$�,.p#UY,!w'�rUVa

¤ '*/�/E¥ � SXlR=\�TWX���f2\�[i = 1, 2

���9gMiTWV=X�`�Y*TdodV=_aR=qkt�o&\�`�Z=u}TWX����9g�YIhaq�\�[]_rTWV=` �x5 �����E5���� �"!$#�%;\�R

GiY*T�ha_aR=X�t x� { YIR<� x { YIR<�|�9gmiTWV=Xl_a[�tm_! lX�tlN

tFu=Z=ZH\kt]Xxha_aR=X�t xzyw{ YIR<� x � { \�f�YIhaq�\�[]_rTWV=` � �"������;YI[]XxX&^�X�o&u}TWX�� Nf"�V=XlR��}TWV=Xxt]\�hau}TW_a\�R�[]X&TWu=[]R=X���_PtC = V1 ∪X(M2)

��_rTWV |C| 6 n/2 + 2m2N���\�`��=_aR=_aR=q�X&^�Z=[]X�tWt]_a\�R�f2\�[

C��_rTWV X&^�Z=[]X�t]t]_a\�R<�~���jYIR<�

TBY��F_aR=q;_aR9TW\�Y�olo&\�u=R9T�haXl`;`bY y�YIR<��TWV=Xwf"Y�o T�TWV�Y*Tm2 6 m

�=TWV=Xxf2\�haha\(��_aR=q;V=\�hP�=t0

c2 � OLVC =|C|

τ(G)6

n2 + 2m2

m6

n

2m+ 2 6

∆ + 1

2+ 2 =

∆ + 5

2

�mO F��tFu=Z=ZH\kt]X�R=\(�eTWV�Y*T�ha_aR=X�t x�� { YIR<� x�� { \�f1YIhaq�\�[]_rTWV=` � �"������ YI[]X�X&^�X�o&u}TWX���_aR<t TWX(Y�� N�"�V=XlR�� |V2| 6 n/2YIR<� TWV=Xbt]\�hau}TW_a\�R[]X&TWu=[]R=X��_PtwTWV=Xb\�R=X�\�f�ha_aR=X x� { ��_�N X�N!�

C = X(M1) ∪ V2N:�pR TWV=_Pt�o�Y�t]X|YIhPtm\DTWV=X

YI[]q�u=`;XlR9Tdt�Z=[]X&i�_a\�u<t]hrg �}X&ikXlha\�ZHX�� V=\�hP� N s�XlR<o&X���X&^�Z=[]X�t]t]_a\�R �mO F��;YIhr��Ylg}t�q�_rikX�t;TWV=XDo&\�`�ZHX&TW_rTW_ri�_rT�g[dY*TW_a\�Y�odV=_aX&ikX�� N�1X&T�u<t�R=\(� t]V=\(� TWV�Y*T�TWV=X�YIR�YIhrg}t]_PtjYI�H\(ikX _PtSY�tmg�`;Z}TW\�TW_Po�YIhahrg�TW_aq�V9T�Nj��\�R<t]_P�}Xl[jY;q�[dYIZ=V

G(V,E)o&\�hahaX�o TW_a\�R \�fRt TBYI[Wt�HX(Y�odV6\�f `�Y*^�_a`�u=`��}Xlq�[]XlX

∆N���\�R<tm_P�}Xl[�TWV=X t]u=�=q�[dYIZ=V

G1\�f

Go&\�R<t]_Pt TW_aR=qb\�f

Y|tmX&T�\�fn/2

X&^�ZH\kt]X���ikXl[mTW_Po&X�tx��_rTWV�[]X�t]ZHX�o T�TW\6Y�`�Y*^�_a`�u=` `�Y*TdodV=_aR=qM\�f

GN ��Xl`�YI[L�6TWV�Y*T

Mo&\�R9TBYI_aR<t \�R=X�X��}q�X�ZHXl[ t TBYI[�YIR<��TWV�Y*TV (G1)

_Pt�YxtmX&T�\�fE_Ptm\�h:Y*TWX��;ikXl[mTW_Po&X�t \�f t]_! lX�R=\�T�h:YI[]q�Xl[¡TWV�YIRn/2N

tFX&TG2 = G[V \ V (G1)]

YIR<��Y�tWtmu=`�XjTWV�Y*TG_Pt�[]X&ikX(YIhaX���ZHXl[�o&hau<t TWXl[Wt

G1YIR<�

G2N�"�V=XlR��

|C| =n

2+ 2R =

n

2+

2n

∆ + 1

�mO(���

τ(G) =n

∆ + 1= R

�mO O��

|C|τ(G)

=∆ + 5

2

�mO C��"�V=_Pt�o&\�`;Z=haX&TWX�t�TWV=XxZ=[]\F\�f�\�f�TWV=XxTWV=Xl\�[]Xl`�N

O��

� �:3 �P3 ¢ A ���&%(7F'I�;L�)9CE7 . /�!"�l/ A )=%(7�0�CH7F'(%*!5+�7��tFu=Z=ZH\kt]XjR=\(� TWV�Y*T�o&hau<tmTWXl[Wt

GiYI[][]_rikXS��_rTWV�R=\�_Ptm\�h:Y*TWX���ikXl[mTW_Po&X�tlN �1X&T

ni�HX�TWV=XS\�[W�}Xl[�\�f

GiN SXlR=\�TWX

�9gMi

�i = 1, 2

� Y;`�Y*^�_a`�u=`�`�Y*TdodV=_aR=qb\�fGi�H�9g

PiTWV=X�t]X&TS\�f�TWV=X X&^�ZH\kt]X��6ikXl[mTW_Po&X�t���_rTWVD[]X�tmZHX�o T

\�fMi

�=�9gpi_rTdt�o�YI[W�}_aR�YIha_rT�g6YIR<��o&\�R<tm_P�}Xl[�TWV=Xwf2\�haha\(��_aR=q;\�R}n~ha_aR=X�YIhaq�\�[]_rTWV=`�N

��� %�!$#�� � � �������� xn, ε

{ �xzyw{ �%� � v � ε > 1 �x � {

C← ∅ ��wv &�&�� # v �$��� G1�

x�� {M1 ← MAX

�MATCHING(G1) �x�� { ! *

n1 6 n/ε� � � #

C← C ∪ V1 �x� { � � � �C← C ∪ X(M1) ��wv &�&�� # v �$��� G2

�x {

M2 ← MAX�MATCHING(G2) �x�� {

C← C ∪ X(M2) �x�� {A2 ← {v ∈ V2 \ X(M2) : ∃u ∈ V1 \ C, uv ∈ E} �x�� { �,+��"-'+��

C← C ∪ A2 �� # �/. � � � �������� xn, ε

{ �

� L�7F/H'*7�� � ��� &4m�/MW�UY�b/��r��nG$'UY�b/Xp�,.p@UY�ES'Ufv*3kbp�UV/MW�p�S�W�WX,.Z'/�) ,NUY�R&�$A,.pM$'*!S'UV/�m�Z'/MWXUY,}v�/Xp � UY�b/XW�/1/�j�,.pXUYp ε0 �UY�E/%*!S�W?wr/Mp�U1S'iR$�&EwBUY�E/HWT$I$'UYpH$Q�|UY�E/�nG$'*-�'&�$�iA,}S'*ε2 − 3ε + 1 � p�k)vM� UY�bS'U#UY�E/`v�$�i�nb/UY,NUY,.ZI,NUY��WTS'UY,}$c$Q�

S'*Dwr$�W�,NUY�ri � ���2 465 �������ε0��S wrS�,.&)p�UfS�&:$�n�UY,.i�S'*/96; S'*Dwr$�W�,NUY�ri6,.p#qT$'kE&�mr/�m Srq�$�Z'/1qX�

2+((∆+1)/ε0)a

¤ '*/�/E¥ � tFX&T ��f2\�[i = 1, 2

�mi = |Mi|

YIR<��R=\�TWX�TWV=X�f2\�haha\(��_aR=q�f"Y�o T�TWV�Y*T�o�YIRb�HX�_a`;`;X��}_:Y*TWXlhrg��}X��}u<o&X��f2[]\�` YIhaq�\�[]_rTWV=` � � �"������=N�1)�+�% > � �UV=XlR=X&ikXl[ ha_aR=X x�� { _Pt�X&^�X�o&u}TWX�����TWV=XlR ha_aR=X x� { o&\�`;Z=u}TWX�t

A2 = ∅ g�TWV=Xl[]Xlf2\�[]X�TWV=X��=R�YIho&\(ikXl[]_aR=qCo&\�`;Z=u}TWX��|_aR�ha_aR=X x�� { tdY*TW_PtQ�=X�t

C = V1 ∪X(M2)N

�1X&TSu<t@�=[WtmTjt]u=Z=ZH\ktmX�TWV�Y*TSha_aR=X x�� { _Pt�X&^�X�o&u}TWX�� N�"�V=XlR��Eu<t]_aR=qbf"Y�o TxME�EX&^�Z=[]X�tWtm_a\�R �~����YIR<�6haXl`;`bY�yx���Xxq�X&T 0

|C|τ(G)

=n1 + 2m2

m6

+ 2m2

m6

m+ 2 6

nεn

∆+1

+ 2 =∆ + 1

ε+ 2

�~y����

�1X&T�u<t R=\(� tmu=Z=ZH\kt]X�TWV�Y*Tbha_aR=X x� { _Pt;X&^�X�o&u}TWX��e_aR<t TWX(Y�� N "�V=XlR���TWV=XDt]X&TC�=R�YIhahrgeo&\�`;Z=u}TWX���_aR

ha_aR=X x� { �9g�YIhaq�\�[]_rTWV=` � � �������� ikXl[]_!�=X�tM0

|C| = |X (M1)|+ |X (M2)|+ |A2| = 2 (m1 + m2) + |A2|�~y�OI�

SXlR=\�TWX �9gQ1 ⊆ V1 \ X(M1)

TWV=X tmX&T�\�fjikXl[mTW_Po&X�tb\�fV1TWV�Y*TbV�Y�tbXlR9TBYI_ahaX��eTWV=XD_aR9TW[]\��}u<o TW_a\�R \�f

t]X&TA2_aR

C�<YIR<�b�9g

B(Q1, A2, EB)��TWV=Xxt]u=�=q�[dYIZ=V|\�f

G_aR<�}u<o&X����9g

Q1 ∪A2NftF_aR<o&XjTWV=X&gbYI[]Xj�H\�TWV

_aR<�}XlZHXlR<�}XlR9T1�"tmu=�<t]X&Tdt \�fP1YIR<�

P2�F[]X�tmZHX�o TW_rikXlhrg)���

B_Pt¡�=_aZ�YI[mTW_rTWX�N SXlR=\�TWXjYIhPtm\ �9g

MBYx`�Y*^�_a`�u=`

`�Y*TdodV=_aR=q;\�fBYIR<��tmX&T

mB = |MB|N tF_aR<o&X

M1 ∪M2 ∪MB_Pt�Y�`�Y*TdodV=_aR=q;f2\�[

G�

τ(G) > m1 + m2 + mB�~y�y��

��\�R<tm_P�}Xl[¡tmX&TX(MB)∩Q1

gk\��9i�_a\�u<t]hrg�� |X(MB)∩Q1| = mBN tF_aR<o&X

G1_Pt�t]u=Z=ZH\kt]X�����_rTWV=\�u}T _Pt]\�h:Y*TWX��

ikXl[mTW_Po&X�t��YIR9g;ikXl[mTWX&^�\�fX(MB)∩Q1

V�Y�t�Y*T�`;\kt T∆− 1

R=Xl_aq�VF�H\�[Wt�_aRA2N�Q�R�TWV=XS\�TWV=Xl[�V�YIR<���

MB�HXl_aR=q�`bY*^�_a`�u=` f2\�[B�<YIR9g�ikXl[mTWX&^b\�f

A2[]X�o&Xl_rikX�t�X��}q�X�t�f2[]\�`�Y*T�haX(Y�t T�\�R=X�ikXl[mTWX&^b\�f

X(MB)∩Q1N

tF\G�|A2| 6 mB(∆− 1)

�~y�M��

O�O

{�hPtm\G�=t]_aR<o&XG1YIR<�

G2YI[]XS�H\�TWV�Y�t]t]u=`;X�����_rTWV=\�u}T�_Ptm\�h:Y*TWX��bikXl[mTW_Po&X�tM��YIZ=Z=ha_Po�Y*TW_a\�R|\�f�haXl`;`�Y y�q�_rikX�tM0

m1 >n1

∆+1

m2 >n2

∆+1

�~y � �

��\�`��=_aR=_aR=qDX&^�Z=[]X�t]t]_a\�R<t%�~y�OI�����~y�y����y�~y�M���YIR<�K�~y � ����ZHXl[]f2\�[]`;_aR=q tm\�`�X;ha_rT]TWhaX|YIR<� X(Y�tmg�YIhaq�Xl�=[dY YIR<�TBY��F_aR=q;_aR9TW\�Y�olo&\�u=R9T

n1 + n2 = n�=\�R=Xwq�X&Tdt0

|C|τ(G)

62 (m1 + m2) + |A2|

m1 + m2 + mB= 2 +

|A2| − 2mB

m1 + m2 + mB6 2 +

∆−3∆−1 |A2|n

∆+1 + |A2|∆−1

�~y����

��X�o�YIhah�TWV�Y*T���X�YI[]Xwo&u=[][]XlR9TWhrg6o&\�R<tm_P�}Xl[]_aR=qbo�Y�tmXn1 > n/ε

�=_�N X�N!�

n2 6 n− n

ε= n

ε− 1

ε

�~y'F��

�Stm_aR=q�X&^�Z=[]X�t]t]_a\�R �~y � ��f2\�[m2

���}XlR=\�TW_aR=q �9gp2TWV=X�RFu=`��HXl[;\�f�TWV=X|X&^�ZH\kt]X�� ikXl[mTW_Po&X�t;\�f

V2��_rTWV

[]X�t]ZHX�o T�TW\M2

�HYIR<��u<t]_aR=q;X&^�Z=[]X�tWtm_a\�R �~y'F����=��Xx\��}TBYI_aR

|A2| 6 p2 = n2 − 2m2 6 n2 −2n2

∆ + 1=

∆− 1

∆ + 1n2 6

(∆− 1)(ε− 1)

(∆ + 1)εn

�~y����

��Xl`�YI[L�wYIhPtm\�TWV�Y*T�X&^�Z=[]X�tWtm_a\�R%�~y���� _Pt�_aR<o&[]X(Y�t]_aR=q���_rTWV |A2|N tF\G��o&\�`��=_aR=_aR=q�X&^�Z=[]X�tWtm_a\�R<t7�~y�����YIR<�|�~y�����0

|C|τ(G)

6 2 +

(∆−3)(ε−1)(∆+1)ε n

n∆+1 + (ε−1)n

ε(∆+1)

= 2 +(∆− 3)(ε− 1)

2ε− 1

�~y'O��

� \�TWX�TWV�Y*T �Ff2\�[�Y��}^�X��ε�(∆−3)(ε−1)/(2ε−1) 6 (∆+1)(ε−1)/(2ε−1)

YIR<� TWV�Y*T�X&^�Z=[]X�tWtm_a\�R��~y����_Ptx�}X�o&[]X(Y�t]_aR=q|��_rTWV

ε� ��V=_ahaX X&^�Z=[]X�tWtm_a\�Rh�~y'O���_Ptx_aR<o&[]X(Y�t]_aR=q=N "�V=X�tmX�T���\�X&^�Z=[]X�tWtm_a\�R<t�Y�t g�`�Z}TW\�TW_Po�YIhahrg

o&\�_aR<o&_P�}Xw��V=XlR

∆ + 1

ε+ 2 = 2 +

(∆ + 1)(ε− 1)

2ε− 1⇐⇒ ε2 − 3ε + 1 = 0

ε>1=⇒ ε0 w 2.62

�~y'C��

tFX&T]TW_aR=qε0 = 2.62

�*��X q�X&TcC2 � OLVC 6 (∆+6.24)/2.62

N�"�V=_Pt�If2\�[�h:YI[]q�X¡i�YIhau=X�t�\�f∆�I_Pt�Y�tmg�`;Z}TW\�TW_Po�YIhahrg

X �9u�YIh�TW\∆/2.62

N

� �:> ' / K 7F' $ /H� . 0��;¥~/�'�%*L�7D+k/H���¡79%*!:%(!2C�!:% � '*)}%*!2/� �:> �5, ¢ )��l7 t > 2

� XwZ=[]\(i�_P�}X�_aR�TWV=_Pt�t]X�o TW_a\�R Y o&\�`;ZHX&TW_rTW_ri�_rT�g�u=Z=ZHXl[��H\�u=R<����q�_rikXlRD_aR|TWV=Xl\�[]Xl` FE��f2\�[�TWV=X o�Y�tmXw��V=Xl[]XTWV=XwRFu=`��HXl[�\�f�o&hau<tmTWXl[Wt�R=XlX��}X���f2\�[�TWV=Xw[]X&ikX(YIha_aR=q�\�f�TWV=Xx��V=\�haXwq�[dYIZ=V6_Pt

O(log n√

n)N "�V=XwZ=[]\F\�f�\�f

TWV=XxTWV=Xl\�[]Xl` �HXl_aR=q|�9u=_rTWXxTWX�odV=R=_Po�YIh}��_rT�_Pt�q�_rikXlR�_aRDYIZ=ZHXlR<�}_r^EN� L�7F/H'*7�� ��� � �E/M&

t = O(log n√

n) � S�&4m S'&G� v*-kEp�UV/MW`,.p%&4$�&�(Q/Mi@n�UY� � &�$ $�&�(+*-,.&�/ S'*Dw�$'W�,NUY��i�M$�W 96; v�S�&`S�vM��,}/MZ'/#v�$�i�nb/UY,NUY,.ZI,NUY��WTS'UY,}$dp�i�S'*.*�/MW�UY�ES�&

∆−2SIw�S',.&GpXU S'&`$�n�UY,.i�S'*4$87�(+*-,.&�/#S'*Dwr$�WX,NUY��i �/MZ'/X&2, �#UY�E/J,.&�n�kbU}(Yw'WTS�n��H,.pJS�UYWT/�/AS�&4m

n,.pDEI&4$�) &2,.& S�m�Z'S�&4v�/Ia

� �:> �P3 '�!5��!:%(� / . %(L�7 +k/H���¡79%*!:%(!2C�!:% � ¥~/�'t = 2{St�Z=[]X&i�_a\�u<tmhrg6_aR tmX�o TW_a\�R�M�N�ME�H��X �=[]_aR=q�TW\�TWV=X f2\�[]X�Y q�[dYIZ=V�YIR<� Y;tmTW[dY*TWXlq�g6f2\�[�[]X&ikX(YIha_aR=qb_rTS_aR6T���\

tmTWXlZ<twtmu<odV�TWV�Y*TwX&ikXl[mg \�R}n~ha_aR=X {j� n�YIhaq�\�[]_rTWV=` o�YIR=R=\�T�Y�odV=_aX&ikX�o&\�`�ZHX&TW_rTW_ri�_rT�g [dY*TW_a\��HX&T]TWXl[xTWV�YIR�TWV=X�H\�u=R<�|Z=[]\(i�_P�}X�� N

O(y

� L�7F/H'*7�� � � ( $�Wt = 2

S�&4m\�M$�WBS'*.*∆ > 2 � &4$ S'* wr$�WX,NUY��i v�S�&]S�vM��,}/MZ'/Bv�$�i@nG/UY,NUY,.Z ,NUY� WTS'UY,}$

p�UYWX,}vU}*3�2q�/U}UV/XW UY�ES�&(∆ + 1)/2

�M$�W|S�w'WTS�n��B$Q��iRS j',.i�kEilmr/Vw'WT/�/∆ � /MZ'/X&c, �d,NU�,.p�qX,3nbS'WXUY,NUV/ ) ,NUY�:&�$

,.pM$�*�S'UV/�mAZ�/MWMUY,}v�/Mp � qT$�UY�\v*-kEp�UV/MWXpy�ES'Z�/#UY�E/#pMS�i�/�p�,��I/JS�&�m�S�&`,.pM$�iR$'WVn��r,}vd$Q�@UY�b/�,.&�n�kbU}(Yw'WTS�n���,.p E &4$�) &,.& S�m�Z'S�&4v�/Ia¤ '*/�/E¥ � � _rikXlR YIR _aR9TWXlq�Xl[

kYIR<� T���\|t]X&Tdt

A = {1, . . . |A|} YIR<� B = {1, . . . |B|} t]u<odV TWV�Y*T |B| =k|A| �<��X�t]X&T A×k B = {(aib(i−1)k+j) ∈ A × B, i ∈ {1, . . . , |A|}, j ∈ {1, . . . , k}} N �pR6\�TWV=Xl[���\�[W�=tM�_af

AYIR<�

BYI[]X�ikXl[mTWX&^Fnpt]X&TdtM�ETWV=X q�[dYIZ=V

(A ∪ B,A×k B)o&\�R<t]_Pt TdtS\�f |A| tmTBYI[Wt�\�f t]_! lX k + 1

[]\F\�TWX��_aR|TWV=XjikXl[mTW_Po&X�t�\�f

AN

�1X&T∆ > 2

�HX YA�}^�X��|_aR9TWXlq�Xl[SYIR<�|t]X&T

n = 2∆(∆ + 1)��M����

� Xw�}X�=R=XH = (V1 ∪ V2, E)

��V=Xl[]X�0

• V1 = N11 ∪ N2

1

�"t]X&TdtN1

1

YIR<�N2

1

YI[]X�`�u}TWu�YIhahrgU�}_PtYP \�_aR9TX�b��_rTWV |N11 | = ∆

� |N21 | = ∆2 �

N11 ∩N2

1 = ∅ g

• V2 = N12 ∪ N2

2

�"t]X&TdtN1

2

YIR<�N2

2

YI[]X�`�u}TWu�YIhahrgU�}_PtYP \�_aR9TX�b��_rTWV |N12 | = ∆2 � |N2

2 | = ∆�

N12 ∩N2

2 = ∅ g

• E = (N11 ×∆ N1

2 ) ∪ (N22 ×∆ N2

1 )N

"�V=X;t]\�npo&\�R<t TW[]u<o TWX���q�[dYIZ=VH_Ptw�=_aZ�YI[mTW_rTWX�����_rTWV=\�u}T�_Ptm\�h:Y*TWX���ikXl[mTW_Po&X�t�YIR<� Y�`;_aR=_a`�u=` o�YI[W�}_aR�YIha_rT�g

ikXl[mTWX&^|o&\(ikXl[]_aR=q�\�fH_Pt�\�f�tm_! lX

τ(H) = n/(∆ + 1)N

�1X&T u<t o&\�R<t]_P�}Xl[ YIR \�R}n~ha_aR=X YIhaq�\�[]_rTWV=` ������=N#� X ��_ahah�t]V=\(� TWV�Y*T TWV=Xl[]X�X&^�_PtmTdt�Y �=_aZ�YI[mTW_rTWXq�[dYIZ=V

G(V1, V2, E)_Pt]\�`;\�[]Z=V=_Po�TW\

Hf2\�[���V=_PodV�TWV=X�o&\�`;ZHX&TW_rTW_ri�_rT�g�[dY*TW_a\j\�f ������j_Pt Y*T¡haX(Y�t T

(∆+1)/2��V=XlRV1_Pt�[]X&ikX(YIhaX���_aR�t TWXlZ OSYIR<�

V2_Pt�[]X&ikX(YIhaX��b_aRbt TWXlZ6yFN z \�[]X�Z=[]X�o&_PtmXlhrg��}TWV=XSt]\�hau}TW_a\�R|o&\�`;Z=u}TWX��

�9g �������_aRGo&\�R9TBYI_aR<twY*TxhaX(Y�tmT

n/2ikXl[mTW_Po&X�t����V=Xl[]X

n = |V1| + |V2|Nw{StWtmu=`;X |V1| = n/2

� haX&TN1�HX�TWV=XjtmX&T�\�f�ikXl[mTW_Po&X�t�\�f

V1_aR9TW[]\��}u<o&X���_aR�TWV=Xjt]\�hau}TW_a\�R��9g:�'���� YIR<��t]X&T

n1 = |N1|N SX�=R=XS_aR9TWXlq�Xl[Wt

p, r, kt]u<odV|TWV�Y*T

n1 = p∆ + r��M}OI�

r 6 ∆− 1��Mky��

k = (∆ + 1)− p��M�M��

� \�TWX�TWV�Y*Tk ≥ 0

YIR<��TWV�Y*T �9u<t]_aR=qxX&^�Z=[]X�tWt]_a\�R<t@��M���� YIR<����M�M����9\�R=X�q�X&Tdt�TWV=X�f2\�haha\(��_aR=qwX&^�Z=[]X�tWtm_a\�R2��M � ��gX&^�Z=[]X�tWtm_a\�R ��Mk����_Pt�_a`�`;X��}_:Y*TWXx�9g�TWV=X �}_Pt]o&u<tWtm_a\�R YI�H\(ikX�0

(k + p)∆ =n

2

��M � �

k 6n

2− n1

��Mk���

�pR<�}XlX�����X&^�Z=[]X�tWt]_a\�R ��Mk����\��9i�_a\�u<t]hrg�V=\�hP�=t�_afk = 0

�n1 = n/2

���<\�[k = 1

�n1 < n/2

�BN tFu=Z=ZH\ktmX R=\(�k > 2 > ∆/(∆− 1)

N���XjTWV=XlR�V�YlikX���u<tm_aR=q�X&^�Z=[]X�tWtm_a\�R<t���M}OI��TW\`��M�M���0n

2− n1 = ∆(∆ + 1)− p∆− r > ∆(∆− p) = ∆(k − 1) > k

��M�F��

tFX&TV1Y�t�TWV=XwZ�YI[mTW_rTW_a\�R6\�f�t]X&Tdt

Np1

�Nk

1

YIR<�R1�}��V=Xl[]X

Np1 ⊂ N1

�Nk

1 ∩N1 = ∅ YIR<�{

|Np1 | = p∆

∣Nk1

∣ = k

��Mk���

O�M

tFu<odV6�}X�o&\�`;ZH\kt]_rTW_a\�R6_Pt�ZH\ktWt]_a�=haXw�HX�o�YIu<t]X�\�f�X&^�Z=[]X�tWt]_a\�R ��Mk���BNf"�V=XlR�� |R1| = k(∆− 1)YIR<�b��XxV�YlikX�0

V1 = Np1 ∪Nk

1 ∪R1��M�O��

tFX&T�YIhPt]\V2Y�t�TWV=X|Z�YI[mTW_rTW_a\�R \�f�t]X&Tdt

Np2

�Nk

2

YIR<�R2��V=Xl[]X |Np

2 | = p��YIR<� |Nk

2 | = ∆kNc"�V=XlR

|R2| = p(∆− 1)N��pRDYIhah}�

V2 = Np2 ∪Nk

2 ∪R2��M�C��

� XxR=\(� o&\�R<tm_P�}Xl[�TWV=[]XlX�o�Y�t]X�t�<R�YI`;Xlhrg�� �5_.�k > 0

YIR<�p > 0

���5_a_.�k = 0

YIR<� �5_a_a_.�p = 0

N<\�[ o�Y�tmX2�5_.�R�

k > 0YIR<�

p > 0����u<t]_aR=q6X&^�Z=[]X�tWt]_a\�R ��M � ���

p∆ < ∆(∆ + 1)YIR<����o&\�R<t]X �9u=XlR9TWhrg��

p ≤ ∆N�"�V=XlR��=��Xwt]X&T

a = p− 1YIR<�

b = ∆− pgaYIR<�

bYI[]XxZH\kt]_rTW_rikX�_aR9TWXlq�Xl[Wt�tWY*TW_Pt]f:g�_aR=qG0

{

k(∆− 1) = a + b∆p(∆− 1) = a∆ + b

� � ���

��X&i�_Ptm_rT�X&^�Z=[]X�tWt]_a\�R<t���M�O��¡YIR<�2��M�C�� YIR<��t]X&TR1 = Ra

1 ∪Rb1

�R2 = Ra

2 ∪Rb2

��_rTWV |Ra1| = a

� |Rb1| = b∆

�|Ra

2| = a∆� |Rb

2| = b��V=Xl[]X

aYIR<�

bYI[]X Y�t|_aRcX&^�Z=[]X�t]t]_a\�R � � ����g�o&\�R<t]_P�}Xl[ YIhPtm\ TWV�Y*T

Ra1

�Rb

1

�Ra

2

� YIR<�Rb

2

YI[]X�`�u}TWu�YIhahrg��}_PtYP \�_aR9T�N`<u=[mTWV=Xl[]`;\�[]X���TWV=X�X��}q�X&npt]X&TE\�f

G_Pt o&\�R<t]_P�}Xl[]X�� Y�t�f2\�haha\(��tM0

E = Np2 ×∆ Np

1 ∪ Nk1 ×∆ Nk

2 ∪ Ra1 ×∆ Ra

2 ∪ Rb2 ×∆ Rb

1

N�<\�[G�2��V=_PodV _PtS_Pt]\�`;\�[]Z=V=_Po TW\

H���ETWV=X

\�R}n~ha_aR=X�t]\�hau}TW_a\�R o&\�`;Z=u}TWX��6�9g �������R=X�o&X�tWtWYI[]_ahrgDo&\�R9TBYI_aR<tSt]X&TN1 ∪Nk

2

\�f o�YI[W�}_aR�YIha_rT�g Y*T�haX(Y�tmTn/2N

QSR�TWV=Xw\�TWV=Xl[�V�YIR<���Np

2 ∪Nk1 ∪Ra

1 ∪Rb2

_Pt�Y�`;_aR=_a`�u=` nptm_! lXjikXl[mTWX&^|o&\(ikXl[lN<\�[�o�Y�tmX\�5_a_.�1�

k = 0����X&^�Z=[]X�t]t]_a\�R ��M�M���q�_rikX�t

p = ∆ + 1YIR<��X&^�Z=[]X�t]t]_a\�R ��Mk����_a`;Z=ha_aX�t

n1 = n/2N

"�V=XlRE_Pt��}X�=R=X��6t]\�TWV�Y*T

G_Pt�_Pt]\�`;\�[]Z=V=_PoxTW\

HN

<\�[jo�Y�tmX%�5_a_a_.�J�p = 0

���E�9g�X&^�Z=[]X�tWtm_a\�R<tR��Mky���YIR<� ��M�F����n/2 − n1 > ∆(∆ + 1) −∆ > ∆

N#tF\���Xo�YIR��}X�=R=X

N11

�N2

1

�N1

2

YIR<�N2

2

Y�t�_aRH

�"t]XlXjTWV=XjTWV=[]XlXx_rTWXl`�t�Y*T�TWV=Xj�HXlq�_aR=R=_aR=q;\�f�TWV=XxZ=[]\F\�f�����_rTWVTWV=X�tmu=Z=Z=haXl`;XlR9TBYI[mg6o&\�R<�}_rTW_a\�R�TWV�Y*T

N11 ∩N1 = ∅ N�pR�YIhah}\�f<TWV=X�YI�H\(ikX�o�Y�tmX�t�kTWV=X�\�R}n~ha_aR=X�tm\�hau}TW_a\�R�o&\�`;Z=u}TWX����9g ������xo&\�R9TBYI_aR<t�Y*T haX(Y�tmT

n/2ikXl[mTW_Po&X�t&N

QSRTWV=X�\�TWV=Xl[ V�YIR<����TWV=X�o&\�[][]X�t]ZH\�R<�}_aR=q q�[dYIZ=V<tG�HXl_aR=q�_Ptm\�`;\�[]Z=V=_Po�TW\

H�

τ(G) = n/(∆ + 1)N

��\�R<tmX �9u=XlR9TWhrg��}TWV=XSo&\�`;ZHX&TW_rTW_ri�_rT�g�[dY*TW_a\�\�f ������ _Pt�Y*T�haX(Y�tmT(∆+1)/2

�}YIR<�;TWV=X�Z=[]\F\�f�\�f TWV=X�TWV=Xl\�[]Xl`_Pt�o&\�`;Z=haX&TWX�N

� \�TWX�TWV�Y*Tj_af���X�YIhaha\(� _Ptm\�h:Y*TWX��DikXl[mTW_Po&X�t�_aRG�H\�R=X�o�YIR X(Y�tm_ahrgDt]V=\(��TWV�Y*Tj\�R=X�o�YIR=R=\�Txq�u�YI[dYIR9TWXlX

o&\�`;ZHX&TW_rTW_ri�_rT�g|[dY*TW_a\�tmTW[]_Po TWhrg|�HX&T]TWXl[�TWV�YIR1/∆

N

� � �p�~"� �p�w� ��� �����&�9� �����p�w��pR�TWV=_Pt�tmX�o TW_a\�RD��X�tm\�`�X&��V�Y*TSodV�YIR=q�X�TWV=Xw��\�[L�F_aR=q�V9g�ZH\�TWV=X�t]X�tjY��}\�Z}TWX�� YIR<�6t]u=Z=ZH\ktmX TWV�Y*T�\�R=X�o�YIRq�\6��Y�o�� \(ikXl[�TWV=X�t]\�hau}TW_a\�Ro&\�R<t TW[]u<o TWX���}u=[]_aR=qDZ=[]X&i�_a\�u<t tmTWXlZ<tlN\� X�Y�tWt]u=`;X;TWV�Y*T�\�R=Xbo�YIR�odV�YIR=q�XTWV=_Pt�tm\�hau}TW_a\�R6�=u}T�tmV=X>�*V=X�V�Y�t�TW\;Z�Ylg|t]\�`;Xwo&\ktmT�f2\�[��}\�_aR=q�_rT�NQSu=[�\�R}n~ha_aR=X�YIhaq�\�[]_rTWV=` f2\�[ TWV=X�o�Y�t]X�\�f�TWV=X���Y�o��9TW[dY�o��F_aR=q�_Pt���Y�tm_Po�YIhahrg YIhaq�\�[]_rTWV=` � �"������=N�"�V=Xt]Z=_a[]_rT�\�f�\�u=[xTWV=\�u=q�V9T Z=[]\�o&X�tWt o�YIR �HX;\�u}TWha_aR=X���Y�twf2\�haha\(��t&N\"�V=X;�HX�tmT�YIZ=Z=[]\(^�_a`�Y*TW_a\�R [dY*TW_a\`�FR=\(��R

f2\�[#{S� _Ptx�H\�u=R<�}X���YI�H\(ikX;�9g�y��2TWV=_Ptw[dY*TW_a\�_PtjX �9u�YIh¡TW\2 − (log log n/ log n)

�L; yI=N�L�BN�QSR TWV=X \�TWV=Xl[V�YIR<������{j� �HXl_aR=qDo&\�`;Z=u}TBY*TW_a\�R�YIhahrg V�YI[W�}Xl[��_rT�_Pt�Y|Z=[]_a\�[]_ ��\�[Wt]XbYIZ=Z=[]\(^�_a`�Y*TWX���TWV�YIR {j��N�tF\G��\�R=Xo�YIR���[]X�tmTW[dYI_aR��|V=Xl[��*V=_a`�t]Xlhaf�_aR�tmX(YI[WodV=_aR=q�f2\�[;o&\�`;ZHX&TW_rTW_ri�_rT�g [dY*TW_a\kt�Y�t�R=X(YI[�Y�t�ZH\ktWt]_a�=haX�TW\�yFN "�V=X`�Y*^�_a`�u=` `�Y*TdodV=_aR=qbZHXl[]f2\�[]`�X��D\�R X(Y�odV�o&hau<tmTWXl[j\�f

G�9g YIhaq�\�[]_rTWV=` � �"������b\��9i�_a\�u<t]hrgDq�u�YI[dYIR9TWXlX�tYIZ=Z=[]\(^�_a`�Y*TW_a\�R [dY*TW_a\ y \�R YIR9geo&hau<tmTWXl[lN "�V=X6f"Y�o T�TWV�Y*T�TWV=X6��V=\�haXDo&\�`;ZHX&TW_rTW_ri�_rT�g�[dY*TW_a\�_Pt��=R�YIhahrg

� �}X&TWXl[]_a\�[dY*TWX���� _Pt��}u=XwTW\;TWV=XwikXl[mTW_Po&X�t�\�f�TWV=X�q�[dYIZ=VBTWV�Y*TSV�YlikXwTW\��HXwTBY���XlR6_aR9TW\bY�olo&\�u=R9TS_aR6\�[W�}Xl[

TW\|o&\(ikXl[wo&[]\ktWtmn~X��}q�X�tM��_�N X�N!��X��}q�X�tx�HX&T���XlXlR�o&hau<tmTWXl[WtlN tF\�TWV=X;YIhaq�\�[]_rTWV=` ��X�Z=[]\�ZH\ktmX _aR ��V�Y*Twf2\�haha\(��ttmTBYI[mTdt���_rTWV�[]u=R=R=_aR=q �E5���� �x5 ��� ��! #�% \�R�X(Y�odVeo&hau<tmTWXl[bYIR<� �9g �}Xlh:Ylg�_aR=q _rTdt;�}X�o&_Pt]_a\�Re\�R TWV=X6o&[]\ktWt nX��}q�X�tJ�5_aR�\�TWV=Xl[���\�[W�=tM�}TWV=XxX&^�ZH\ktmX���ikXl[mTW_Po&X�t�\�f�YIR9gbo&hau<tmTWXl[�YI[]X#�=[Wt TWhrg�o&\�R<t]_P�}Xl[]X��6Y�t�R=\�T��HXlha\�R=q�_aR=qTW\|TWV=X�tm\�hau}TW_a\�R�u=R<�}Xl[ o&\�R<t TW[]u<o TW_a\�R��BN � X&^FT ��\�R<o&XbYIhah o&hau<t TWXl[Wt�[]X&ikX(YIhaX�����q�[dYIZ=V

GB_Ptwf2\�[]`;X�� YIR<�

�x5 ��� �x5���� ��! #�% x � { _Pt�[]u=R�N "�V=X7�=R�YIh<t]\�hau}TW_a\�R _Pt�TWV=X�u=R=_a\�R \�f<TWV=X�XlR<�}ZH\�_aR9Tdt \�f YIhah}TWV=X�X��}q�X�t []X&TBYI_aR=X��

O �

�9gDTWV=X;t]u<olo&X�t]t]_rikX;[]u=R<tx\�f �E5������x5 ��� �"! #�%=N��pR ��V�Y*Txf2\�haha\(��tM��f2\�[V ′ ⊆ V

����X �}XlR=\�TWX �9gG[V ′]

TWV=Xt]u=�=q�[dYIZ=V�\�f

G_aR<�}u<o&X����9g

V ′ N��� %�!$#�� � � ��������%�

C← ∅ �*%���i← 1

��t � � C← C ∪ X(MAX � MATCHING(Gi)) � �

B← G[∪ti=1(Vi \ X(MAX�MATCHING(Gi)))] ��,+��"-'+��

C← C ∪ X(MAX � MATCHING(B)) �� # � � � � ��������%�

�1X&T u<t¡�}XlR=\�TWX��9gκTWV=X�o&\ktmT��}u=X�TW\jTWV=X�odV�YIR=q�X�\�f�TWV=X�tmTBY*TWu<t¡\�f�YjR=\�R}npo&\(ikXl[]_aR=q�ikXl[mTWX&^�TW\ Yxo&\(ikXl[]_aR=q

\�R=X�N�{�hPtm\G��Y�t�Z=[]X&i�_a\�u<t]hrg��¡��Xb�}XlR=\�TWX|�9gMiTWV=X�X��}q�X&npt]X&T;o&\�`;Z=u}TWX���9g �E5���� �x5 ��� ��! #�% x

Gi{ YIR<�

�9gmi_rTdt;o�YI[W�}_aR�YIha_rT�g��

i = 1, . . . , tg�f2u=[mTWV=Xl[]`;\�[]X�����X��}XlR=\�TWX|�9g

MBTWV=X|X��}q�X&nptmX&Tbo&\�`;Z=u}TWX�� �9g

�x5 ��� �x5���� ��! #�% xB{ YIR<�|�9g

mB_rTdt�o�YI[W�}_aR�YIha_rT�gkN

� L�7F/H'*7�� £ � 0 �E/@v�$�i�nb/UY,NUY,.ZI,NUY�1WTS'UY,}$d$Q� S�* wr$�WX,NUY��i � � (2 465 �1S wrS�,.&GpXU S�&H$�n�UY,.iRS'*�$ 7 (+*-,.&�/�S'*Dwr$�W�,NUY�ri�M$�W 96; ,.pdqT$'kE&�mr/�m\Srq�$�Z'/AqX�

2κa

¤ '*/�/E¥ � {St�\�R=X o�YIR�tmXlX f2[]\�`sYIhaq�\�[]_rTWV=` � � �"������G��TWV=X�ikXl[mTW_Po&X�t odV�YIR=q�X�� YI[]X TWV=X \�R=X�t�\�f�TWV=Xt]X&T ∪ti=1(Vi \ X(Mi))

N {�`;\�R=qbTWV=X�t]X�ikXl[mTW_Po&X�t��X&^=Y�o TWhrg |X(MB)| = 2mBikXl[mTW_Po&X�txZ�Y�tWtxf2[]\�` R=\�R}n

o&\(ikXl[]_aR=q6TW\Do&\(ikXl[]_aR=qD\�R=X�tlN`tFu=Z=ZH\kt]X�TWV�Y*T f2\�[�X(Y�odV \�f�TWV=Xl` Y�o&\kt TκV�Y�twTW\6�HX�Z�YI_P� N%"�V=XlR���TWV=X

o&\�`;ZHX&TW_rTW_ri�_rT�g|[dY*TW_a\;\�f � � �"������;o�YIR��HXx��[]_rT]TWXlR �5u<t]_aR=q�X&^�Z=[]X�t]t]_a\�R �}O��L��Y�t

cBt � OLVC =

(

2t∑

i=1mi

)

+ 2κmB

(

t∑

i=1mi

)

+ mB

6 2κ� � OI�

YIR<��TWV=XwZ=[]\F\�f�\�f�TWV=XjTWV=Xl\�[]Xl`�_Pt�o&\�`;Z=haX&TWX�N"�V=Xl\�[]Xl` O|�}[dYl��txTWV=X�q�XlR=Xl[dYIh�o�Y�t]X ��V=Xl[]X�R=\�f2u=[mTWV=Xl[ t]ZHX�o&_!�<o�Y*TW_a\�R\�f�TWV=X;T�g�ZHX;\�f�TWV=X�o&\ktmT�_Pt

q�_rikXlR�N � XxR=\(��Y�tWtmu=`;XxT���\;o&\ktmT]n~`;\��}XlhPt0

• κ_Pt�YA�}^�X���o&\�R<tmTBYIR9T g

• κ_Pt�Y*T�`;\ktmT�X �9u�YIh�TW\ TWV=XxTW\�TBYIh1RFu=`��HXl[�\�f�ikXl[mTW_Po&X�t�odV�YIR=q�X�� N

<\�[�TWV=X1�=[WtmT�o&\kt T]n~`�\��}Xlh}�<u<tm_aR=q�X&^�Z=[]X�t]t]_a\�R � � OI���=TWV=X�f2\�haha\(��_aR=q;[]X�t]u=hrT�_Pt��}_a[]X�o TWhrg|Z=[]\(ikX�� N

¢ /�'*/ A5A )<' � � � +.�#S ��jr/�m|v�$�pXU �bS�p�UV$|q�/7nGS�,}m��M$'WJS�&)� Z'/MWMUV/Tj�(VpXUVS'UYkEp�iR$Im�, �fv�S'UY,}$�& � UY�E/M&:UY�E/�v�$'i@nG/MU}(,NUY,.ZI,NUY��WTS'UY,}$\$Q�JS�* wr$�WX,NUY��i � ���2 465 �|S wrS�,.&GpXU7S�& $�n�UY,.iRS'* $87 (+*-,.&4/ 9 ; (QS'*Dwr$�WX,NUY��i ,.pJv�$'&GpXUVS�&)UVa�1X&Txu<tSR=\(� f2\�o&u<tj\�u=[WtmXlhrikX�tw\�RDTWV=X t]X�o&\�R<� \�f¡TWV=X o&\ktmT]n~`;\��}XlhPtxt]ZHX�o&_!�=X���YI�H\(ikX�Nd=\�[STWV=_Ptxo�Y�t]X�� ��Xt]\�`;X&��V�Y*T�`;\��}_af:g|YIhaq�\�[]_rTWV=` � � �"�������Y�t�tmV=\(��RAP u<tmT��HXlha\(�J��YIR<�DY�t]t]u=`;X n

�FR=\(��R�_aR6Y���i�YIR<o&X�N��� %�!$#��$� � ��������%��wv &�&�� # v �$��� G1

�xzyw{

C← X(MAX�MATCHING(G1)) ��wv &�&�� # v �$��� Gi, i = 2 . . . , t

�x � { *%���

i← 2��

t � �x�� {Mi ← MAX

�MATCHING(Gi) �x�� {

Pi ← Vi \ X(Mi) �x� {B← G[∪ij=1Pj] �

O(�

x {MB ← MAX

�MATCHING(B) �x�� { !!*

mB 6√n

� � � #C← C ∪ X(Mi) �x�� { � � � �C← C ∪ Vi �x�� { *"!

xzy � { � �xzy y {i0 ← � v � � ���"�'&��"�(��) ����� �%� x�� { ��� � � ��)������� �xzy � {B′ ← G[∪i0j=1Pj] �xzy � { �,+��"-'+��

C← C ∪ X(MB′) �� # � �$� � ��������%�

� L�7F/H'*7���� � +.�#�M$�W�S�&)�cZ�/MWMUV/�j vM�ES�&Ewr/�m � UY�b/�v�$'p�UA$Q��UY�E/�vM�ES'&xwr/�/��Mk)S'*-p�UY�E/:UV$'UVS'* &)kEi�q�/XW�$Q�Z'/MWXUY,}v�/Xp\vM�ES�&Ewr/�m � UY�E/M& UY�E/%v�$�i�nb/UY,NUY,.ZI,NUY�:WTS'UY,}$B$L�|S'*Dw�$'W�,NUY��i � � (2 4 5 �:SIw�S',.&GpXU#S�&<$Xn�UY,.iRS'*7$87 (+*-,.&4/96;�(QS'*Dw�$'W�,NUY��i ,.p�qT$�kE&4m�/�m\SrqT$'Z�/ q��

3√

na

¤ '*/�/E¥ � ��Xl`�YI[L�6�=[WtmT�TWV�Y*T�TWV=Xc\�R=hrg ikXl[mTWX&^FnpodV�YIR=q�X�t ZHXl[]f2\�[]`;X��v�9g YIhaq�\�[]_rTWV=` � � ���'���� YI[]X\�RX(MB′)

�2��V=Xl[]XB′ _Pt�TWV=X�q�[dYIZ=Veo&\�R<tmTW[]u<o TWX�� _aR ha_aR=X x y � { ��YIR<����f2u=[mTWV=Xl[]`;\�[]X�� TWV�Y*T mB′

YIhrn��Ylg}t�tWY*TW_PtL�=X�t

mB′ 6√

n� � y��

�pf ha_aR=X x�� { \�f1YIhaq�\�[]_rTWV=` � � ���'�����_Pt�R=\�T�X&^�X�o&u}TWX���Y*T�YIhah}��_�N X�N!�F_af i0 = t�5ha_aR=X xzy y { ���FTWV=XlR

mB′ 6√

nN

��\�R<tmX �9u=XlR9TWhrg��Hu<tm_aR=q;X&^�Z=[]X�tWt]_a\�R<t�� � OI��YIR<�c� � y���0

cMt � OLVC 6 2mB′ 6 2√

n� � M��

QSR�TWV=Xw\�TWV=Xl[�V�YIR<����t]u=Z=ZH\ktmXwTWV�Y*T�ha_aR=X x�� { _Pt�X&^�X�o&u}TWX�� Y*T�haX(Y�tmT�\�R<o&X�Nf"�V=XlR��

mB >√

n� ��� �

�Stm_aR=q�X&^�Z=[]X�t]t]_a\�RK� � y�����tmX&T]TW_aR=qκ = mB′

YIR<���}XlR=\�TW_aR=q6�9gv(C)

TWV=X�i�YIhau=X;\�f�TWV=X�tmX&TC�5_aR

v(C)YIR9gbikXl[mTWX&^�R=\�R6odV�YIR=q�X��6o&\�u=R9Tdt�O�YIR<��YIR9g�ikXl[mTWX&^|odV�YIR=q�X��6o&\�u=R9Tdt��}��Xxq�X&T 0

v(C) 6 n− 2mB′ + 2κmB′ 6 n + 2m2B′

� � ��� SXlR=\�TWX��9g

mTWV=X�o�YI[W�}_aR�YIha_rT�g \�fEYS`�Y*^�_a`�u=` `�Y*TdodV=_aR=qj\�f

GYIR<��u<t]X�X&^�Z=[]X�tWtm_a\�R<t��}O������ ��� ��YIR<�%� � ���BN

"�V=XlR��}TWV=Xwf2\�haha\(��_aR=q;V=\�hP�=t�f2\�[�TWV=X�o&\�`;ZHX&TW_rTW_ri�_rT�g|[dY*TW_a\;\�f � � �"������G0

cMt � OLVC =v(C)

τ(G)6

n + 2m2B′

m6

n

mB+ 2mB′ 6 3

√n

� � F��

� ^�Z=[]X�t]t]_a\�R<t�� � M���YIR<�c� � F���o&\�R<o&hau<�}XwTWV=XlR|TWV=XwZ=[]\F\�f�\�f�TWV=XwTWV=Xl\�[]Xl`�N� X6R=\(� t]V=\(� TWV�Y*T�TWV=X6[]X�t]u=hrTb\�fSTWV=Xl\�[]Xl` C�_Pt��9u=_rTWX6TW_aq�V9T ��t]_aR<o&XDR=\ \�R}n~ha_aR=X YIhaq�\�[]_rTWV=` o�YIR

Y�odV=_aX&ikX o&\�`;ZHX&TW_rTW_ri�_rT�g|[dY*TW_a\:��YIq9YI_aR<tmTSYIR�\�Z}TW_a`�YIh1\�9�n~ha_aR=X�\�R=X>���HX&T]TWXl[�TWV�YIRO(√

n)N

� L�7F/H'*7�� ,�¦ � � &4m�/MWRUY�E/R���XnG$'UY�E/MpM/MpR$Q�AUY�E/�$�WT/Mi � &�$:$'&)(+*-,.&�/HS'* wr$�WX,NUY��i �M$'W 96;]v�S�& S�vM��,}/MZ'/ �S wrS�,.&)p�U S'&�$�n�UY,.iRS�*4$87�(+*3,.&4/�S�* wr$�WX,NUY��i � v�$�i@nG/UY,NUY,.ZI,NUY�dWTS'UY,}$ √2n � /MZ�/M&\, �yUY�E/y,.&�n�kbU}(Yw'WTS�n���,.p�qX,3nGS�WMUY,NUV/) ,NUY�E$�kGUf,.pM$'*!S'UV/�m�Z�/MWMUY,}v�/MpJS�&4mn,.pFE &4$�) &2,.& S�m�Z'S�&4v�/Ia

¤ '*/�/E¥ � �1X&T(∆, n1) ∈ N× N

YIR<��t]X&Tn = (1 + ∆)n1

N�{�T�TWV=XJ�=[WtmTStmTWXlZ��V1_PtSYIR6_aR<�}XlZHXlR<�}XlR9Tjt]X&T

\�f�tm_! lXn1N%�pf�Z=h:YlgkXl[ y|tmXlhaX�o Tdt�tm\�`�X ikXl[mTW_Po&X�t \�f

V1��TWV=XlR�TWV=X���V=\�haX�_aR<tmTBYIR<o&X�_Pt�Y|q�[dYIZ=V ��_rTWV=\�u}T

YIR9g6X��}q�X�N@�pR6TWV=_PtSo�Y�t]X��HTWV=X�\�Z}TW_a`�YIh�i�YIhau=Xτ∗(G)

_Ptj�E�H��V=Xl[]X(Y�t�TWV=X \�R}n~ha_aR=X i�YIhau=X�_Pt�ZH\kt]_rTW_rikX�N�"�V=X[]X�t]u=hrTW_aR=q;[dY*TW_a\�X �9u�YIhPt ∞ YIR<��TWV=XxTWV=Xl\�[]Xl`�V=\�hP�=t&N

O F

��\�R<tmX �9u=XlR9TWhrg�����X;o�YIR f2\�o&u<t�\�u=[WtmXlhrikX�t�\�R�TWV=X�o�Y�t]X;��V=Xl[]X;Z=h:YlgkXl[�y�t]XlhaX�o Tdt�R=\|ikXl[mTW_Po&X�t �}u=[]_aR=qTWV=X��=[Wt Txt TWXlZ�N��pRDTWV=_PtSo�Y�tmX��ETWV=X�_aR<t TBYIR<o&X�q�[dYIZ=V o&\�R<t]_PtmTdt�\�f

n1t TBYI[WtS\�f¡t]_! lX

(1 + ∆)[]\F\�TWX�� _aR

V1�

\�R=X�t TBYI[�ZHXl[�ikXl[mTWX&^�_aRV1N�"�V=XlR��=TWV=Xw\�Z}TW_a`�YIh�i�YIhau=X�tWY*TW_PtL�=X�td�5[]X�o�YIhah1TWV�Y*T

n = (1 + ∆)n1�

τ∗(G) =n

∆ + 1= n1

� � ���

SXlR=\�TWX6�9gV ′

1

TWV=X6t]X&T�\�fSikXl[mTW_Po&X�t�\�fV1TWV�Y*T�YI[]XDodV�YIR=q�X���_aRe\�[W�}Xl[ TW\ �HX�_aR<o&hau<�}X��e_aR TWV=X��=R�YIh

t]\�hau}TW_a\�R �5_�N X�N!� TWV=X ikXl[mTW_Po&X�tj_aR9TW[]\��}u<o&X�� _aRDTWV=X t]\�hau}TW_a\�R�YIf:TWXl[STWV=X���Y�o��9TW[dY�o��F_aR=qx�BN�"�V=XlR��1tm\�hau}TW_a\�RCo�YIR��HXx��[]_rT]TWXlRDY�t

C = V ′1 ∪ V2 \ Γ(V ′

1)f2\�[�Y�TW\�TBYIh�o&\ktmT�\�f

v(C) =∣

∣V ′1

2+ ∆

(

n1 −∣

∣V ′1

) � � O����\�R<tmX �9u=XlR9TWhrg��HZ=h:YlgkXl[Sy�odV=\F\kt]X�t�HY*T��HX�tmT �HY�t]X&T

V ′1

\�f�o�YI[W�}_aR�YIha_rT�g

β∗ ∈ {�[]q�`;_aRλ∈[0, n

∆+1 ]

{

β2 −∆β +∆n

∆ + 1

} � � C��

SX�=R=X∆ = 2n1

N�QSR=Xwo�YIR6X(Y�t]_ahrg|t]V=\(� TWV�Y*T�X&^�Z=[]X�tWtm_a\�R � � C���_a`;Z=ha_aX�tβ∗ = n1

�EYIR<��o&\�`��=_aR�Y*TW_a\�R6\�fX&^�Z=[]X�tWtm_a\�R<t�� � ����YIR<�c� � O�����_rTWV

β∗ = n1_aR<�}u<o&X�t�o&\�`;ZHX&TW_rTW_ri�_rT�g|[dY*TW_a\�tdY*TW_Ptmf:g�_aR=q

v(C)

τ∗(G)= n1 =

−1 +√

1 + 8n

4>√

2n�~�����

��V=Xl[]XxTWV=X�h:Y�tmT�_aR=X �9u�YIha_rT�g�_aR6X&^�Z=[]X�tWtm_a\�R �~������V=\�hP�=t��HX�o�YIu<tmX f2u=R<o TW_a\�R √· �HXl_aR=q�o&\�R<o�YlikX�� (−1/4) +√

(1 + 8n)/2 > (−1/4) + ((√

1 +√

8n)/2) >√

2nN�"�V=_Pt�o&\�`;Z=haX&TWX�t�TWV=X�Z=[]\F\�f�\�f�TWV=XxTWV=Xl\�[]Xl`�N

� � �&�K ���~������l�K�~� �� x� �H�����1�*�|� �w��������w������ X�o&\�R<t]_P�}Xl[ _aR�TWV=_Pt¡tmX�o TW_a\�R�YIR \�R}n~ha_aR=X�`;\��}Xlh=t]u=Z=ZH\ktm_aR=qwTWV�Y*T TWV=X�_aR=Z=u}T]n~q�[dYIZ=V;_Pt []X&ikX(YIhaX�� �9g�`;X(YIR<t\�f�_rTdt�X��}q�X�tlN�"�V=X&g6YI[][]_rikX \�R=X�Y*TxY�TW_a`;X�N =\�[�X&ikXl[mg�R=X&� X��}q�X��HTWV=X�ha_aRG��t�\�f�_rTdt�XlR<�}ZH\�_aR9Tdt���_rTWV6TWV=XXlR<�}ZH\�_aR9Tdt�\�fxTWV=X X��}q�X�t�YIha[]X(Y���g Z=[]X�tmXlR9TDYI[]X�YIhPtm\ []X&ikX(YIhaX�� N ��X tmu=Z=ZH\kt]X TWV�Y*T |E| _Pt|�FR=\(��RU_aRY���i�YIR<o&X�����X�tmX&T

E = {e1, . . . , e|E|}�H��V=Xl[]X

eiYI[]X�RFu=`��HXl[]X��D_aR \�[W�}Xl[x\�f TWV=Xl_a[wYI[][]_ri�YIh}��YIR<� �}X&i�_PtmX

TWV=Xwf2\�haha\(��_aR=qbYIhaq�\�[]_rTWV=`�N��� %�!$#�� � �"�������

t← 1 �C← X(et) �*%���

t← 2���� � � � � !!*

X(et) ∩ C = ∅ ��� � #C← C ∪ X(et)

*"!�� ��,+��"-'+�� � �� # �/. � � �"�������

{Stw\�R=X�o�YIRtmXlX�f2[]\�` TWV=XbYIhaq�\�[]_rTWV=` YI�H\(ikX���TWV=X�_a[][]X&ik\�o�YI�=_aha_rT�g�_aR�TWV=Xbo&\�R<t TW[]u<o TW_a\�R\�f�TWV=X�\�R}n~ha_aR=Xt]\�hau}TW_a\�R��}X(YIhPtj��_rTWV TWV=X XlR<�}ZH\�_aR9Tdtw\�f�YIR X��}q�X�Y�twY���V=\�haX�Nd�U_rTWV []X�t]ZHX�o TxTW\6Y�`;\��}Xlh ��Y�tmX�� u=ZH\�RYI[][]_ri�YIh�\�f�ikXl[mTW_Po&X�t�_rT�_Pt�Y�t�\�R=X YIhaha\(��t�=f2\�[�X&ikXl[mg|X��}q�X�YI[][]_ri�_aR=qG� Y���Y�o��9TW[dY�o��F_aR=q�\�f�haX&ikXlh1\�R=X�N¤ '*/H� /E��!:%*!2/ . 3 � 0 �E/Rv�$�i@nG/UY,NUY,.Z ,NUY�HWTS�UY,}$�$Q��S�* wr$�WX,NUY��i��(2 4 5 �\S wrS�,.&)p�UyS�& $Xn�UY,.iRS'* $ 7 (+*-,.&�/�S'* wr$'(WX,NUY��i ,.pJq�$�kE&4m�/�mHSrqT$�Z'/ qX�R_Ea 0 �r,.pdqT$'kE&�m�,.p#UY,!w'�rUVa¤ '*/�/E¥ � {St]t]u=`;X�TWV�Y*TwTWV=X !!*9n~_aR<t TW[]u<o TW_a\�R \�f�YIhaq�\�[]_rTWV=` � �"������|_PtjX&^�X�o&u}TWX��

qTW_a`�X�t&NA�~Tx_PtjX(Y�tmgDTW\

t]XlX�TWV�Y*T�TWV=XqX��}q�X�t�XlR9TBYI_aha_aR=q�TWV=X�t]XSX&^�X�o&u}TW_a\�R<t�f2\�[]`vY�`�Y*^�_a`�YIhH`�Y*TdodV=_aR=q�\�f

GN "�V=XSt]X&T

C�=R�YIhahrg

O(�

o&\�`;Z=u}TWX��D�9g6TWV=X YIhaq�\�[]_rTWV=` ikXl[]_!�=X�t |C| = 2qN�Q�RDTWV=X�\�TWV=Xl[SV�YIR<�����9g6X&^�Z=[]X�t]t]_a\�R<�~�����

τ(G) > qN

"�V=Xwo&\�`;ZHX&TW_rTW_ri�_rT�g|�H\�u=R<�Dy�_Pt�TWV=XlR�_a`;`;X��}_:Y*TWXlhrg|�}X��}u<o&X�� N��\�R<tm_P�}Xl[ Y�tmTBYI[�[]X&ikX(YIhaX���X��}q�X&n~�9gFn~X��}q�X�N¡{�haq�\�[]_rTWV=` � �����������_ahah�_aR9TW[]\��}u<o&X�_aR

CTWV=X�XlR<�}ZH\�_aR9Tdt�\�f

TWV=X��=[WtmT X��}q�X�[]X&ikX(YIhaX���YIR<� R=\wR=X&�eikXl[mTWX&^���_ahah<�HX�_aR9TW[]\��}u<o&X���_aRCh:Y*TWXl[lN "�V=X�\�Z}TW_a`�YIh=ikXl[mTWX&^Fnpo&\(ikXl[

f2\�[�YIR9g|t TBYI[�o&\�R<t]_Pt Tdt�\�f�_rTdt�o&XlR9TWXl[lN~tF\;V=Xl[]X��=TWV=Xw�H\�u=R<�Dy�_Pt�Y*T]TBYI_aR=X�� N�~T�_Pt�X(Y�tmg TW\�tmXlX6TWV�Y*T�TWV=XD\�R}n~ha_aR=XD`;\��}Xlh P u<tmTb�}X�tWo&[]_a�HX��e_Pt�X �9u=_ri�YIhaXlR9TbTW\�TWV=X6\�R=X���V=Xl[]X YIhah

ikXl[mTW_Po&X�txYI[]X Z=[]X�t]XlR9Txf2[]\�`�TWV=X��HXlq�_aR=R=_aR=q�\�f�TWV=X q9YI`�X�YIR<�6X��}q�X�txYI[]X Z=[]X�t]XlR9TWX��D\�R=X&n~�9gFn~\�R=X�N#s�Xl[]X����V=XlR=X&ikXl[�YIR�X��}q�XbYI[][]_rikX�t�R=\�R=X�\�f�TWV=X;XlR<�}ZH\�_aR9Tdt�\�f���V=_PodV�YI[]X�_aR

C��TWV=XlR �H\�TWV�\�f�_rTdtwXlR<�}ZH\�_aR9Tdt

XlR9TWXl[CN

� ���r�=�K���&������QSR}n~ha_aR=X�o&\�`;Z=u}TBY*TW_a\�R _Pt¡Y�o TWu�YIhahrg;Y�ikXl[mg Y�o TW_rikX�YI[]X(YS\�f<TWV=X�TWV=Xl\�[]X&TW_Po�YIh�o&\�`�Z=u}TWXl[ tWo&_aXlR<o&X�YIR<��h:YI[]q�Xlhrg_aR9TWXl[]X�tmTdt TWV=X�\�ZHXl[dY*TW_a\�R�YIhE[]X�tmX(YI[WodV=Xl[Wt�f2[]\�` �H\�TWV�TWV=Xl\�[]X&TW_Po�YIh YIR<��Z=[dY�o TW_Po�YIhHZH\�_aR9Tdt�\�fEi�_aX&�1��t]_aR<o&X�TWV=X`�Y*TWV=Xl`�Y*TW_Po�YIh�Z=[]\��=haXl`�t�V=Xl[]X�Xl`�YIR�Y*TWX f2[]\�` `;\��}XlhPt�X&^�Z=[]X�t]t]_aR=q�[]X(YIha_rT�g�`�\�[]X�[]_PodV=hrg�TWV�YIR6TWV=X�o&\�R}nikXlR9TW_a\�R�YIhE\�R=X�tlN "�V=X�ikXl[mTWX&^Fnpo&\(ikXl[]_aR=q Z=[]\��=haXl` �}X(YIhrT�_aR�TWV=_Pt Z�YIZHXl[�_Pt \�R=X�\�f TWV=X�o&XlR9TW[dYIhEZ=[]\��=haXl`�t _aRo&\�`��=_aR�Y*TW\�[]_:YIhE\�Z}TW_a`;_! (Y*TW_a\�Rb_aR�_rTdt \�9Hn~ha_aR=X�ikXl[Wtm_a\�R�N {�t¡��X�V�YlikXSt]XlXlR�Y*T TWV=X��HXlq�_aR=R=_aR=q�\�fETWV=X�Z�YIZHXl[�_rT�[]Xl`�YI_aR<t�ikXl[mg�R�Y*TWu=[dYIh X&ikXlR�_aRb_rTdt�\�R}n~ha_aR=XjikXl[Wtm_a\�R�N "�V=Xl[]XSX&^�_PtmTdt�Y�RFu=`��HXl[�\�f1\�ZHXlR�Z=[]\��=haXl`�t�TWV�Y*Tt]XlXl` _aR9TWXl[]X�t TW_aR=qDf2\�[wf2u=[mTWV=Xl[ t TWu<�}_aX�tlN|�_a[WtmTw\�f�YIhah}�1TWV=X;_a`;Z=[]\(ikXl`;XlR9T ��_af�ZH\ktWtm_a�=haX���\�f TWV=X�o&\�`;ZHX&TW_rnTW_ri�_rT�g�[dY*TW_a\kt�\��}TBYI_aR=X��YIR<��TWV=XbY�odV=_aX&ikXl`;XlR9T�\�f�ha\(��Xl[��H\�u=R<�=t�f2\�[wTWV=X�o�Y�t]X ��V=Xl[]X

t < nNb{�hPt]\G��Y

f2u=[mTWV=Xl[�q�XlR=Xl[dYIha_! (Y*TW_a\�Rb\�f TWV=X�ikXl[mTWX&^Fnpo&\(ikXl[]_aR=q�_Pt¡TWV=X�\�R=X���V=Xl[]X���X�o&\�R<t]_P�}Xl[ ��Xl_aq�V9Tdt�\�R�TWV=X�ikXl[mTW_Po&X�t\�f�TWV=X|_aR=Z=u}T]n~q�[dYIZ=VUYIR<���X�tmX(YI[WodVef2\�[�Y `;_aR=_a`�u=` TW\�TBYIhrn���Xl_aq�V9T�ikXl[mTWX&^�o&\(ikXl[lN �pR TWV=X6o&\�`;Z�YIR9g`;\��}XlhEZ=[]X�t]XlR9TWX��|_aR�t]X�o TW_a\�R�O'��TWV=_Pt�q�XlR=Xl[dYIha_! (Y*TW_a\�R6V�Y�t�Y�ikXl[mg�R�Y*TWu=[dYIh�_aR9TWXl[]Z=[]X&TBY*TW_a\�R�_af1��Xjo&\�R<t]_P�}Xl[TWV�Y*T�TWV=Xx`�YIRFu=f"Y�o TWu=[]_aR=q;\�f�YIR|\�[W�}Xl[�V�Y�t�_rTdt�Z=[]\�ZHXl[�o&\ktmT�YIR<��TWV=Xwo&\�`;Z�YIR9g���_Pt]V=X�t�TW\�`;_aR=_a`�_! lX�TWV=Xo&\ktmTj\�f�TWV=X�`�YIRFu=f"Y�o TWu=[]_aR=q|_aR t]u=�Eo&\�R9TW[dY�o TW_aR=q=N���Xl[]f2\�[]` Y�t]XlR<tm_rTW_rikX;YIR�YIhrg}t]_Ptj\�f \�R}n~ha_aR=X YIhaq�\�[]_rTWV=`�tf2\�[�TWV=_Pt���Xl_aq�V9TWX���ikXl[Wt]_a\�R6\�f���{j� tmXlXl`�t�TW\;u<t�Y ikXl[mg|_aR9TWXl[]X�t TW_aR=q�\�ZHXlR�Z=[]\��=haXl`�N

� ��� �l�9���r�}���;rOX= � N¡{�u<t]_aXlhaha\G� � N�<Xlu=Xl[Wt TWXl_aR���tHN¡�1Xl\�R�YI[W�}_}���¡Nft9TW\�u=q�_aX���YIR<� z N�"�YIh:YI`;\=N{�haq�\�[]_rTWV=`�t f2\�[�TWV=X\�R}n~ha_aR=XwTW[dYlikXlha_aR=q�tWYIhaX�t]`�YIR�Z=[]\��=haXl`6N��@* wr$�WX,NUY��i�,}v�S��Ey'CG� � ��0z�'F�� � �'O}O'�1y����}O�N

; yI= �wN ��YI[mn�� XlVFu<�<YbYIR<�BtHN � ikXlR�NS{ ha\�o�YIhrn~[dY*TW_a\�TWV=Xl\�[]Xl` f2\�[jYIZ=Z=[]\(^�_a`�Y*TW_aR=qbTWV=X ��Xl_aq�V9TWX��6ikXl[mTWX&^o&\(ikXl[�Z=[]\��=haXl`6N��y&G&4a��#,.pMvMWT/UV/� nrn�*�a��S'UY�ba!�Hy��x0zy�� � � FE��O C�Ok�FN

; M>=��SN ��Xl[]q�X�N �~WTS�n��rp1S'&�m����XnG/MW?w'WTS�n���pBN � \�[mTWV`s�\�hah:YIR<����{�`�t TWXl[W�<YI`��1O Ck��M�N; � = z N" SXl`�YIR=q�X�����N�� YI[dY��}\�R�� YIR<�%{�N�"xN�� Y�t]odV=\ktlN�Q�R}n~ha_aR=X `�Y*^�_a`�u=` n~\�[W�}Xl[�_aR<�}u<o&X���V=Xl[]X��}_rTBYI[mgt]u=�=q�[dYIZ=V Z=[]\��=haXl`�t&N �pR {�N�s�h:Yli����'����N � N���X9EXl[mg���YIR<���=N �U_aX��}Xl[]`�YIR=R�� X��}_rTW\�[Wt����� ( ����_���� ��! 0 �E/�$'W��|S�&�m " WTS�vUY,}v�/J$Q� +�& �M$'W�iRS�UY,}vXp��Fik\�hau=`;X�O C�F�M�\�f C ;����FZ�YIq�X�t�Mky'F � M�M � N�tFZ=[]_aR=q�Xl[mn{ Xl[]h:YIqG�Ey�������N

; �I= �wN � hrn���YIR=_ri4�H{ N)�_:Y*T � �wN � N���YI[]Z��EYIR<� � N�"�u=[]Z=_aR�N���\�`;ZHX&TW_rTW_rikX�YIR�YIhrg}t]_Pt�\�f��=R�YIR<o&_:YIh�q9YI`;X�t&N�pR#" WT$IvIa ( � ;���$ '_��<Z�YIq�X�t�Mky�� � M�M�ME��O C�CkyFN

;DF>= { N��_:Y*TjYIR<� � N)�&%wq�_aR=q�Xl[�HX��}_rTW\�[Wt&N'� &)*-,.&4/ S'*Dwr$�WX,NUY��i�p)(1UY�E/dp�UVS�UV/ $Q�1UY�E/ S�WMUY�<ik\�hau=`�X�O ��� y \�f C ;��<N~tFZ=[]_aR=q�Xl[mn}{ Xl[]h:YIqG��O C�C�O�N

; �I= { N � g*�I[]f+��t YIR<�,�}NI�1XlV=Xlh�N�QSR=ha_aR=X�YIR<�J�=[WtmT]n?�}T¡o&\�ha\�[]_aR=qkt�\�f=q�[dYIZ=V<tlN.-Ea��~WTS�n�� 0 �E/�$ra!�=O(yx0zy�O(� � y�y��x�O C�O�O�N;DO>= z N z NfsSYIhahP��/�[WtWtm\�RUYIR<� z N~tx lXlq�X���gkN �1\(��Xl[;�H\�u=R<�=t�f2\�[;\�R}n~ha_aR=X6q�[dYIZ=V o&\�ha\�[]_aR=q=N 0 �E/�$�W�/MUVa

;�$�i@n�kbUVa���vX,}a!��O�M��G�mOI��0PO F�M � O(� � ��O C�C � N

O O

;DC>=AtHN)�p[dYIR=_�YIR<��{�N �wN ��YI[]ha_aR�N QSR=ha_aR=Xwo&\�`;Z=u}TBY*TW_a\�R�N��pR �N�tHNGs�\�odVF��YIu=`��=X��}_rTW\�[� � nrn�WT$ j',.iRS'UY,}$'&S'*Dwr$�WX,NUY��i�p@�M$'W C "~(V�ES�WTm|n�WT$xqX*!/Mi�p�� odV�YIZ}TWXl[|O�ME�¡Z�YIq�X�t;��y�O � �'Fk�FN � � t �¡u=�=ha_PtmV=_aR=q���\�`;Z�YIR9g��O C�Ck�FN

;rO��>= �¡N��1\(i���tL �� z N�t}Y���tM�¡YIR<� � N�"�[]\�T]TWXl[lN6{�R�\�R}n~ha_aR=X�q�[dYIZ=Vo&\�ha\�[]_aR=q�YIhaq�\�[]_rTWV=` ��_rTWV�t]u=�=ha_aR=X(YI[ZHXl[]f2\�[]`�YIR<o&X�[dY*TW_a\=N �#,.pMvMWT/UV/ BS'UY�Ea!�E���b�mO n�M���0�M}O C � Mky��x��O C�O�C�N

;rO�OX=��SN�s N�� YIZ�Y��}_a`;_rTW[]_a\�ubYIR<� ��NEt9TWXl_aq�ha_rTT �N ;�$�i�qX,.&4S'UV$�WX,}S'*�$�n�UY,.i�,��IS'UY,}$�& (�S'*Dwr$�W�,NUY�ri�p�S�&4mAv�$�i@n�*!/�j'(,NUY��N��¡[]XlR9TW_Po&XdsSYIhah}� � X&� ��Xl[WtmX&g���O C�O}O�N

;rO(yI=, �NxtFhaX(Y*TW\�[�YIR<� �xN � N�"�YI[L (YIR�NE{�`;\�[mTW_! lX���X � o&_aXlR<o g�\�fHha_PtmT¡u=ZE�<Y*TWX�YIR<��Z�YIq�_aR=qx[]u=haX�tlN/;�$�i�i�kb&�a� ;�`�Ey'OG�~y���0zy��ky � y���OE��O C�Ok�FN

O C

� � �k���w��x�F! ����k�� �"�V=X;`bYI_aR�_P�}X(Y�\�f�TWV=X�Z=[]\F\�f�_Pt�YIR�YIha\�q�\�u<t�TW\�TWV=X�\�R=X�\�f�TWV=X�Z=[]\F\�f�\�f�_rTWXl` yb_aR TWV=Xl\�[]Xl` yFN�"�V=XX&^FTW[dY;�}_ � o&u=hrT�g�V=Xl[]X�_Pt��HX�o�YIu<t]X \�f

t = o(n)N7�~TStmV=\�u=hP�6�HX�R=\�TWX���TWV�Y*T�TWV=X�f"Y�o T

t = n_aR�TWV=Xl\�[]Xl` y

_a`;ZH\�[mTBYIR9TWhrg|t]_a`;Z=ha_!�=X�t�TWV=Xw[]X�tmZHX�o TW_rikX Z=[]\F\�f�N{Stj_aR�t]X�o TW_a\�R M�N�ME�1��X;o&\�R<tm_P�}Xl[ Y�T���\�n~Z=h:YlgkXl[wq9YI`;X�N �¡h:YlgkXl[;O�[]X&ikX(YIhPtjTWV=X _aR<tmTBYIR<o&X;�9g o&hau<tmTWXl[Wt

��V=_ahaXDZ=h:YlgkXl[�y�o&\�R<t TW[]u<o TdtbTWV=X t]\�hau}TW_a\�R���_�N X�N!��_rTbZ�YI[mTW_rTW_a\�R<tVi_aR9TW\ T���\�t]u=�<tmX&Tdt0

SiYIR<�

Ci��TWV=X

f2\�[]`;Xl[�\�R=Xj�}XlR=\�TW_aR=q�TWV=XxtmX&T�\�f1TWV=XS_aR<�}XlZHXlR<�}XlR9T�ikXl[mTW_Po&X�t�\�fViYIR<��TWV=Xjh:Y*T]TWXl[�\�R=Xx�}XlR=\�TW_aR=q�TWV=Xjt]X&T

\�f TWV=X o&\(ikXl[]_aR=qbikXl[mTW_Po&X�tj\�fViN���X �}XlR=\�TWX��9g -"�r5�� � TWV=X�Z=[]\�o&X��}u=[]X��1[]XlZ=[]X�t]XlR9TW_aR=q�o&\�R<tmTW[]u<o TW_a\�R�\�f

TWV=X��}X�o&_Ptm_a\�R \�f¡Z=h:YlgkXl[wyx�1YI�H\�u}TSTWV=X Z�YI[mTW_rTW_a\�R \�fViN��pRD\�TWV=Xl[���\�[W�=t�HTWV=X��}X�o&_Pt]_a\�R \�f¡Z=h:YlgkXl[wy ��_ahah

�HX;�}XlR=\�TWX����9g(Ci, Si)← PLAY2(Vi)

N z \�[]Xl\(ikXl[����X;o�YIhah�q�S�mb*3�2v�$�Z'/MWT/�m Ydn~haX(YlikX�t�tmTBYI[j��V=\kt]X�YIhah

haX(YlikX�t�YI[]X�_aR<o&hau<�}X�� _aR�TWV=X�ikXl[mTWX&^�o&\(ikXl[�u=R<�}Xl[�o&\�R<tmTW[]u<o TW_a\�R�N`"�V=X;[]X&ikX(YIha_aR=q tmTW[dY*TWXlq�g�\�f�Z=h:YlgkXl[bO_Ptw��Y�tmX�� u=ZH\�R�TWV=X�f2\�haha\(��_aR=q6`;\��}u=haX��}XlR=\�TWX�� �9g � # � �, � � � � ��YIR<� �}X(YIha_aR=q���_rTWV�TWV=X;[]X&ikX(YIha_aR=qD\�fY�t]_aR=q�haX�ikXl[mTWX&^EN��~Tw_Ptxo�YIhahaX�� ��_rTWV�_aR=Z=u}Tdt

∆ ∈ N�G[∪j=1,...,iVi]

�1_�N X�N!�1TWV=X;q�[dYIZ=V YIha[]X(Y���g�[]X&ikX(YIhaX����YIR<��T���\�t]X&Tdt�\�f�ikXl[mTW_Po&X�t

RSYIR<�

RCg}_rT�[]X&TWu=[]R<t�Y�R=X&� ikXl[mTWX&^

y��_rTdt�ha_aRG��t���_rTWV�TWV=X�ikXl[mTW_Po&X�t�YIha[]X(Y���g

[]X&ikX(YIhaX�� YIR<� TWV=X�tmX&TdtRSYIR<�

RCu=ZE�<Y*TWX�� NB"�V=X�R=Xl_aq�VF�H\�[Wt�\�f

xo&\�R<tm_P�}Xl[]X�� _aR TWV=X�TWV=_a[W��!!*�\�f

YIhaq�\�[]_rTWV=` � # � �� � ��� � ���}X(YIh ��_rTWV�TWV=Xxq�[dYIZ=VDYIha[]X(Y���g�[]X&ikX(YIhaX�� N <\�[�[]X(Y�t]\�R<t�\�f�t]_a`;Z=ha_Po&_rT�g��}��Xxt]X&T�_aR��V�Y*T�f2\�haha\(��t

G[∪j=1,...,iVi] = [G]iN

��� %�!$#���� # � �, � � � � �(�! * |RC| = ∆

��� � #�� ���$)'��������) �%�'� �� v � � #��&��� � ��� |RC| �� � � � !!* |RS| 6= ∅ ��� � # )!������'� x ∈ RS �� ��� ��!�����'�����%� � � � ��� � �! * � � v � ∆ ����� �������'&%� ��� � #RS ← RS \ x

*�!� � � � � ���$�"�'��� v �%�'� �*�!

� # � ��� # � �, � � � � �(�

�pRbTWV=Xj\(ikXl[dYIhah�[]X&ikX(YIha_aR=q�YIhaq�\�[]_rTWV=`��}��Xju<tmXwYIR6Y��=�}_rTW_a\�R�YIh�`;\��}u=haXxo�YIhahaX��UPDATE(X1, Y1, X2, Y2)

YIR<�t]ZHX�o&_!�=X��|_aR|��V�Y*T�f2\�haha\(��t&N��� %�!$#��!+�- � 5�� � �! *

X1 )��,��� v � �(� v � ���'�"� v ��'� #%�'&��%� � ��� � # v ����� � � � X2*�!

! *Y1 )��,��� v � �(� v � ���'�"� v ��'� #%�'&��%� � ��� � # v ����� � � � Y2

*�!� # � �!+�- � 5�� � �

"�V=X#�=R�YIh1q�[dYIZ=VGV�Y�t�Y�f2\�[]` YIR�YIha\�q�\�u<t�TW\ TWV=Xw\�R=Xx\�f �=q�u=[]X�O�N��~T�o&\�R<tm_PtmTdt�\�f+0

• ∆�=ha\�o���t�\�f tm_! lX

∆ + 1�<TWV=X

∆− 1�=[Wt TS\�R=X�t��HXl_aR=qbtmTBYI[WtSYIR<�|TWV=X h:Y�t TS\�R=X �HXl_aR=q1P u<tmTjY TW[]XlX

YIR<�

• Y�[]\F\�T]n�ikXl[mTWX&^�t]u<odV|TWV�Y*T�TWV=Xx��V=\�haXwq�[dYIZ=V�_Pt�Y TW[]XlXw\�f��}Xlq�[]XlX ∆

{St]t]u=`;X∆ ∈ N

��t]X&Tn = ∆(∆ + 1) + 1

YIR<�t = d2∆ log ∆e + (2 + K ′)(1 + ∆) + ∆ + 2

� f2\�[K ′ = d1 + 2 log ∆e ��YIR<�eo&\�R<t]_P�}Xl[�TWV=X6f2\�haha\(��_aR=qt TW[dY*TWXlq�geZ=h:YlgkX��e�9g Z=h:YlgkXl[ O�f2\�[�[]X&ikX(YIha_aR=q TWV=Xq�[dYIZ=V�_aR

ttmTWXlZ<tlN

��� %�!$#�� %�5�� � �xzyw{

K′ ← d1 + 2log∆e �x � {K← ∆− 2K′ − 3 �x�� {S← ∅ �

y��

x�� {C← ∅ �x� {i← 0 �x {A← 0 �� - �r5 � ! � y ��� v �!����� v ���"� �x�� { � �"! � �

A < K∆ � �x�� {i← i + 1 �x�� {Vi ← �!��������,����,�����'�'������������� dK− A/∆e �xzy � {(Ci, Si)← PLAY2(Vi) �xzy y {S← S ∪ Si) �xzy � {C← C ∪ Ci) �xzy � {A← A + ∆|Si|+ |Ci| �xzy � { � �xzy � {

r← A− K∆ �xzy {R1C ←

v � ���(��������� � ����) v &����!� v ���,� � & �xzy � {XS ← S �xzy � {XC ← C \ R1C �xzy � {R1S ← ∅ �x � � {t← d2∆log∆e − i �� - �r5 � ! � � � v �� �(�,� � �,�"���������%� � � � ���'& �������,���� � �x � y { *%���

j← 1�� � � �x � � {

y← ONE�VERTEX(∆, [G]i, R

1S, R

1C) �x � � {

i← i + 1 �x � � {Vi ← {y} �x � � {(Ci, Si)← PLAY2(V

�i) �� � �$��� ���$) v �'� |Vi| = 1

�x � {

S← S ∪ Si �x � � {C← C ∪ Ci �x � � {A← A + ∆|Si|+ |Ci| �x � � { +�- � 5 � � (Si, Ci, R1S, R1C)x�� � { � �� - �r5 � ! � � � v �� �(�,� � �,�"���������%� )'���"��& �%���()�� ��� �Y5 � � ��� ∆ � � ��� )��,� � � v �"� � � � ���'&���� )�� � � ���&%� �x�� y {

R2C ← ∅ �x�� � {R2S ← ∅ �x�� � { *%���

j← 1��

K′(1 + ∆) � �x�� � { !!*A 6 ∆(K + K′)

��� � #s← 1x�� � { � � � �s← 2 �x�� {V1i ← ∅ �x�� � { *"!

x�� � { *���k← s

�� � � �x�� � {y← ONE

�VERTEX(∆, [G]i, R

kS, R

kC) �x�� � {

Vki ← {y} �x�� y { � �x�� � {Vi ← V1i ∪ V2i �x�� � {i← i + 1 �x�� � {(Ci, Si)← PLAY2(V

�i) �� � �$��� ���$) v �'� |Vi| 6 2

�x�� � {

S← S ∪ Si �x�� {C← C ∪ Ci �x�� � {A← A + |Ci ∩ V1i| �

y�O

x�� � { *���k← s

�� � � � +�- � 5�� � (Si ∩ Vki, Ci ∩ Vki, RkS, RkC) � �x�� � {v3 ← |R1C ∪ R2C| �� - �r5 � ! � � � ��&� � R1S, R

2S, R

1C, R

2C �%� RS, RC

�x� � {

RS ← ∅ �x� y {RC ← R1C ∪ R2C �x� � { *%���

j← 1��

2(∆ + 1)− v3 � �x� � {i← i + 1 �x��� {y← ONE

�VERTEX(∆, [G]i, RS, RC) �x� � {

Vi ← {y} �x� {(Ci, Si)← PLAY2(Vi) �x� � {S← S ∪ Si �x� � {C← C ∪ Ci �x� � { +�- � 5 � � (Si, Ci, RS, RC) �x � { � �x y {

R1S ← R1S ∪ R2S �x � { � �"! � �R1S 6= ∅ � �x � {i← i + 1 �x�� {y← ONE

�VERTEX(∆, [G]i, R

1S, ∅) �x � {

Vi ← {y} �x {(Ci, Si)← PLAY2(Vi) �x � {C← C ∪ Ci �x � { � �� - �r5 � ! � � � � v � � �%�"��)!� �x � { ������� ← false �x�� � {

j← 0 �x�� y { � � - � 5��x�� � {

y← ONE�VERTEX(∆, [G]i, RS, RC) �x�� � {

i← i + 1 �x�� � {j← j + 1 �x�� � {Vi ← {y} �x�� {(Ci, Si)← PLAY2(Vi) �x�� � {S← S ∪ Si �x�� � {C← C ∪ Ci �x�� � { +�- � 5�� � (Si, Ci, RS, RC)x�� � { ������� ← (RS 6= R)x�� y { !!*

Ci )��,�"� v � �(� v � �"�'�"� v �%�'� #��&��%� � � � � # v ��� � �!� RC*"!

x�� � { +�#���!!� ������� ���j = 1 + ∆ �x�� � { ! * ������� � � � #

i← i + 1 �x�� � {Vi ← {y} �����'&�� � ���$)��,������),��'� �� � v ��� �%� v � � #%�'&��� � ��� RC �x�� � {(Ci, Si)← PLAY2(V

�i) �x�� {

C← C ∪ Ci �x�� � { *�!x�� � { *%���

k← 1��

∆−j � �x�� � {i← i + 1 �x�� � {Vi ← {y} ���%��&�� � ��� )��,�"�%��),��������%� � �%� � �x�� y {(Ci, Si)← PLAY2(V

�i) �x�� � {

C← C ∪ Ci �x�� � { � �� - �r5 � ! � � )'� � �%�"���(�'�,����� % � � ���%� � v �,��)�� � � ���& �

y�y

x���� {i← i + 1 ��Vi ��� ����������� ��� #�'&����")'��� �%� ��� &���#%� v ���'� �x�� � {Vi ← ∅ �x�� { ���'& v � � x ∈ XS � Vi )��,��� v � �(� ∆ #%�'&�����)���� )��,�"�%��),��������%� � �� � �x�� � { v & � � �"& v &��'� � � v &����,���'����#�'&����")'��� ��� XC �!�"�%� )'� � � �%�'&%� ��� ∆ #��&����")'�� �x�� � { ���'& v � � ��� ���%� )'� �(� �%�'&�� )�&�� v ������!����!�%� x� � { v �������%��#��&��� � �!� Vi �x�� � { ���'& � v )!�$#%�'&��� � # v �"������� ����� �%� x�� � {

��� ���$# � � �������%��#��&����")'������ ����� )'�'&"&��� �%�������!�"� )'� � � �%�'& �xzy � � { ���'& v � � )'��������) �%�'��)'� � ���,�%���"��) �������%���"& v �"� v �'&� v � � &��'#� v �����v �"� �� % v #��&��� � )'�,�"�%�) ���� �� v #��&��%� � ����)���� �����&���� y �xzy � yw{(Ci, Si)← PLAY2(V

�i) �xzy � � {

S← S ∪ Si �xzy � � {C← C ∪ Ci �� # � � %�5�� � �

' 7�� � ) > �[�a ,.&�/Mp ����� UV$ � ����� <������ ��� > $Q� � � � �\UVS,E'/1S'U iR$'p�U d2∆ log ∆e p�UV/on�pa��@U�*-,.&�/ � ����� � K∆ 6

A < (K + 1)∆ � r < ∆ � |C1|/∆ ∈ NS�&�m � iR$'WT/�$�Z'/MW � |S1|+ |C1|/∆ = K

a_Ea�yU *3,.&4/ � ����� <o/M&�m\$L������������ � >I� i = d2∆ log ∆e S�&4m A 6 (K ′ + K)∆

a

¤ '*/�/E¥x/E¥w!:%(7�� , � =[]\�` ha_aR=X x�� { \�R=X q�X&Tdt∆|Si| + |Ci| 6 ∆|Vi| < (K + 1)∆ − A

N s�XlR<o&X���TWV=Xo&u=[][]XlR9T i�YIhau=X�\�f

AtWY*TW_PtL�=X�t

A < (K +1)∆�Ft]\

r < ∆N¡QSR TWV=X�\�TWV=Xl[ V�YIR<����o&haX(YI[]hrg��

A = ∆|S|+ |C| N��\�R<tmX �9u=XlR9TWhrg�� Y*T|TWV=X�XlR<�U\�f�TWV=X � ��!!� � n~ha\F\�Zc_aR ha_aR=X x y � { �S\�R=X�V�Y�t |S| 6 KYIR<���j`�\�[]Xl\(ikXl[�

|C| = q∆ + r��_rTWV

q + |S| = KN tF\G� |C1| = q∆

N"�\�o&\�R<o&hau<�}XjTWV=XSZ=[]\F\�f�\�f1_rTWXl` O'����X�V�YlikX�TW\ tmV=\(�UTWV�Y*T�TWV=X � �"! � � n~ha\F\�Z�\�f1ha_aR=X x�� { _Pt�X&^�X�o&u}TWX��

Y*Tx`�\kt T d2∆ log ∆e TW_a`;X�t&N SXlR=\�TWX �9g AiTWV=X�i�YIhau=X;\�f

AY*TxTWV=X XlR<� \�f¡TWV=X

iTWV X&^�X�o&u}TW_a\�R�\�f TWV=X

ha\F\�Z�N~tFX �9u=XlR<o&X(Ai)i

tWY*TW_PtL�=X�tM� ∀i �<t]u<odV|TWV�Y*T Ai < K∆0

Ai+1 > Ai + K − Ai

�"{ NPOI�

�1X&T�R=\(�Bi = K∆−Ai

N�tFX �9u=XlR<o&X(Bi)i

tWY*TW_PtL�=X�t��5u<tm_aR=q X&^�Z=[]X�t]t]_a\�R��"{�NPOI�L��TWV=X�f2\�haha\(��_aR=q _aR<�}u<o TW_a\�R[]u=haX

{

B0 = K∆Bi+1 6 Bi

(

1− 1∆

)

∀i t]u<odV|TWV�Y*T Bi > 0

�"{ Nzy��

<[]\�` X&^�Z=[]X�t]t]_a\�R`�"{�Nzy�����\�R=X�o�YIR �}X��}u<o&X�TWV�Y*T � ∀i � Bi−1 > 0YIR<�

Bi 6 K∆(1−∆−1)i N <u=[mTWV=Xl[]`;\�[]X��f2[]\�` X&^�Z=[]X�tWt]_a\�R<t��"{ NPOI�L�¡YIR<�2�"{�Nzy����Bi�HX�o&\�`;X�t R=\�R}n~ZH\ktm_rTW_rikXSf2\�[

i > log (K∆)/− log((1−∆−1)i)N

"�V=_Pt�h:Y�tmT��9u�YIR9TW_rT�g _Pt;tm`�YIhahaXl[�TWV�YIR2∆ log ∆

N���\�R<t]X �9u=XlR9TWhrg���TWV=X � ��!!� � n~ha\F\�Z \�f�ha_aR=X x� { _Pt R=\�TX&^�X�o&u}TWX���`;\�[]XjTWV�YIR d2∆ log ∆e TW_a`;X�t�YIR<�bTWV=X�Z=[]\F\�f�\�f�_rTWXl` Oj_Pt�o&\�`;Z=haX&TWX�N¤ '*/�/E¥�/H¥;!:%�7�� 3 � "�V=X�i�YIhau=X�\�f

io&h:YI_a`;X���f2\�haha\(��t _a`�`;X��}_:Y*TWXlhrg f2[]\�` TWV=XbTW\�TBYIh�RFu=`��HXl[�\�f�TWV=X

_rTWXl[dY*TW_a\�R<t \�f�TWV=X *���kn~ha\F\�Z�N�{�T�X(Y�odV _rTWXl[dY*TW_a\�R��A�}\FX�twR=\�T�_aR<o&[]X(Y�t]X��9g `;\�[]X TWV�YIR O�N|"�V=Xl[]Xlf2\�[]X��

u<t]_aR=q6_rTWXl` O'�A < (K + 1)∆ + d2∆ log ∆e 6 (K + K ′)∆

YIR<��TWV=X;Z=[]\F\�f5t�\�f�_rTWXl` y|YIR<��\�f�TWV=XhaXl`;`�Y YI[]Xwo&\�`�Z=haX&TWX�N

<[]\�` YIhaq�\�[]_rTWV=`1� # � �, � ��� � � ��Xbo�YIR �}X��}u<o&XbTWV�Y*T�Y*T�ha_aR=X x� � { ��TWV=Xbq�[dYIZ=V []X&ikX(YIhaX�� o&\�R<tm_PtmTdt\�f |XS|

_Ptm\�h:Y*TWX��D_aR<�}XlZHXlR<�}XlR9TSikXl[mTW_Po&X�t�E\�f |XC |��Y `�u=hrTW_aZ=haX \�f

∆��_Pt]\�h:Y*TWX��Do&\(ikXl[]_aR=q�ikXl[mTW_Po&X�tM� \�f

s2��Y��}hrg o&\(ikXl[]X��tmTBYI[Wt�\�f�tm_! lX∆ + 1

��\�fσ2��Y��}hrg o&\(ikXl[]X��tmTBYI[Wt�\�f�t]_! lX

u2 + 1�u2 < ∆

YIR<� \�fv2_Pt]\�h:Y*TWX���o&\(ikXl[]_aR=q;ikXl[mTW_Po&X�tlN z \�[]Xl\(ikXl[�

σ2 = |RS1 | ∈ {0, 1} YIR<� v2 = |RC

1 | 6 ∆N

y�M

' 7�� � ) � � �yU�*-,.&4/ � ��� � <o/M&�m|$Q� ������� �� � >>� i = K ′(1 + ∆) + d2∆ log ∆e S�&4m A = ∆(K + K ′)a

¤ '*/�/E¥ � "�V=X i�YIhau=X�\�fi_a`;`�X��}_:Y*TWXlhrg�f2\�haha\(��tSf2[]\�` TWV=X TW\�TBYIh�RFu=`��HXl[S\�f _rTWXl[dY*TW_a\�R<tj\�f TWV=X *���kn~ha\F\�Z

�5ha_aR=X�t x� � { TW\ x�� y { �BN�<[]\�` YIhaq�\�[]_rTWV=` � # � �� � ��� � ����_afs = 1

f2[]\�` ha_aR=X x� yw{ TW\ x�� � { ��TWV=XlR TWV=Xq�[dYIZ=V []X&ikX(YIhaX����}u=[]_aR=qDX&ikXl[mg�ha\F\�Z ��_rTWV

k = 1o&\�R<tm_PtmTdt�\�f

λ1��Y��}hrg o&\(ikXl[]X���t TBYI[Wt�\�f�t]_! lX

∆ + 1�

σ′1 6 1

��Y��}hrg�o&\(ikXl[]X�� t TBYI[Wt�\�fHt]_! lX1+u′

1

�u′

1 < ∆YIR<�

v′1_Pt]\�h:Y*TWX�� o&\(ikXl[]_aR=qxikXl[mTW_Po&X�t&N�"�V=X�_aR<o&[]X(Y�t]_aR=q

\�fAo&\�[][]X�tmZH\�R<�=tjTW\�TWV=X�RFu=`��HXl[S\�f¡o&\(ikXl[]_aR=q�ikXl[mTW_Po&X�txY��=�}X��D_aR

C�}u=[]_aR=q�X&^�X�o&u}TW_a\�R \�f¡ha_aR=X�t x�� y {

TW\ x�� � { � YIR<��TWV=_Pt�RFu=`��HXl[�_Pt�X �9u�YIh�TW\K ′∆

N�tF\G�A > ∆K + ∆K ′ Ny�pf s = 2

Y*TSha_aR=X x�� � { �H��X�YIhPt]\V�YlikX

A = (K + K ′)∆YIR<��TWV=XwZ=[]\F\�f�\�f�TWV=XwhaXl`;`�Y _Pt�o&\�`�Z=haX&TWX�N

� X tm\�V�YlikXA = ∆(K + K ′) = |XS|∆ + |XC | + s2∆ + λ1∆ + u′

1 + v′1 ⇒ u′1 + v′1 ∈ {0,∆} N<u=[mTWV=Xl[]`;\�[]X���TWV=X ikXl[mTW_Po&X�t�[]X&ikX(YIhaX����}u=[]_aR=q6X&ikXl[mg ha\F\�Z���_rTWV

k = 2_aR<�}u<o&XbY�t]u=�=q�[dYIZ=V�o&\�R<t]_Pt TW_aR=q

\�fλ2��Y��}hrg�o&\(ikXl[]X���tmTBYI[Wt;\�fxtm_! lX

∆ + 1�

σ′2 6 1

��Y��}hrg�o&\(ikXl[]X��etmTBYI[Wt;\�fjtm_! lX1 + u′

2

�u′

2 < ∆YIR<�v′2_Pt]\�h:Y*TWX�� ikXl[mTW_Po&X�tM����_rTWV

u′2 + v′2 ∈ {0,∆} N1"�V=X�q�[dYIZ=V []X&ikX(YIhaX�� u=R9TW_ah�TWV=X�XlR<� \�f -��x5 � ! � �

�5ha_aR=X x�� � { ���Fo&\�R<t]_Pt Tdt \�f |XS|_Pt]\�h:Y*TWX��;_aR<�}XlZHXlR<�}XlR9T�ikXl[mTW_Po&X�t�F\�f |XC |∆

_Ptm\�h:Y*TWX���o&\(ikXl[]_aR=qwikXl[mTW_Po&X�t�s3��Y��}hrgFnpo&\(ikXl[]X�� t TBYI[Wt�\�fjtm_! lX

∆ + 1��\�f

σ3 6 1��Y��}hrg�o&\(ikXl[]X�� t TBYI[Wt�\�fjtm_! lX

1 + u3�

u3 < ∆��\�f

σ′3 6 1

��Y��}hrg�o&\(ikXl[]X�� tmTBYI[Wtw\�f�t]_! lX1 + u′

3

�u′

3 < ∆��YIR<��\�f

v3 6 2∆_Pt]\�h:Y*TWX�� o&\(ikXl[]_aR=q6ikXl[mTW_Po&X�t&N

<u=[mTWV=Xl[]`;\�[]X��v3+u3+u′

3 ∈ {0,∆, 2∆} YIR<� λ1+λ2+σ′1+σ′

2 = 2K ′ N ��X(YIhahrg�� σ′1 = |R1

S|�σ′

2 = |R2S |YIR<�

v′1 + v′2 = |R1C ∪R2

C | = v3N

' 7�� � ) ��� ( WT$�i *-,.&�/ ������� UV$�*-,.&�/ ����� ��< ����������� � >>� 2(∆ + 1)vM*-kEpXUV/XWXp��ES�Z'/:q�/�/M& WT/MZ�/�S'*!/�m�a

�$�W�/�$�Z'/MW � S�Uf*-,.&�/ ������ � |RC | < ∆a

¤ '*/�/E¥ � "�V=XxhaXl`�`�Y f2\�haha\(��t�_a`;`;X��}_:Y*TWXlhrgbf2[]\�` TWV=Xwf"Y�o T�TWV�Y*T�_afv3 6= 0

�=TWV=XlRv3 = 2∆− v′1 − v′2

N� X�R=\(� �}X�tWo&[]_a�HX�TWV=X�q�[dYIZ=V []X&ikX(YIhaX�� f2[]\�` ha_aR=X x � { TW\�TWV=X�XlR<�D\�f�YIhaq�\�[]_rTWV=` %�5 � � � -��r5 � � �

YIR<�$-��x5 � � �BN �~T6YIhr��Ylg}t|o&\�R9TBYI_aR<t|TWV=X |XS|_Ptm\�h:Y*TWX�� _aR<�}XlZHXlR<�}XlR9T6ikXl[mTW_Po&X�tM�STWV=X |XC |

_Pt]\�h:Y*TWX��o&\(ikXl[]_aR=qDikXl[mTW_Po&X�tMg _rT�YIhPtm\ o&\�R9TBYI_aR<t

2K ′ + 2\�[

2K ′ + 1�"�}XlZHXlR<�}_aR=q \�R ��V=X&TWV=Xl[

RC_Pt�Xl`;Z}T�g�\�[

R=\�TX����Y��}hrg o&\(ikXl[]X���tmTBYI[WtS\�f�t]_! lX∆ + 1

N1�pfRC 6= ∅

�HTWV=XlRRS = {x} YIR<�DTWV=X�q�[dYIZ=V�o&\�R9TBYI_aR<tx\�R=X��Y��}hrg�o&\(ikXl[]X��tmTBYI[� []\F\�TWX�� Y*T

x��\�f�t]_! lX

1 + ∆ − |RC |N ��Xl`�YI[L�B�=R�YIhahrg�TWV�Y*T�f2[]\�` ha_aR=X x � { TW\

TWV=X�XlR<��\�f�YIhaq�\�[]_rTWV=`��∆ + 2

o&hau<t TWXl[Wt YI[]X�[]X&ikX(YIhaX�� N tF\G��TWV=X�TW\�TBYIh�RFu=`��HXl[ \�f�o&hau<tmTWXl[Wt _Pt X&^=Y�o TWhrgN = d2∆ log ∆e+(K ′+2)(∆+1)+∆+2 = O(∆ log ∆)

N =[]\�` ha_aR=X x � { TW\Sha_aR=X x�� � { � - �r5 � ! � � ���Y�TW[]XlX;\�f�tm_! lX

∆ + 1V�Y�tj�HXlXlR�[]X&ikX(YIhaX�����_rTWV�\�R=hrg \�R=X ikXl[mTWX&^ _aR

S�2TWV=_PtxikXl[mTWX&^ _Pt

x�BN "�V=XlR��1TWV=X

o&\�`;Z=haX&TW_a\�R�\�fGY*Twha_aR=X�t x��� { TW\|ha_aR=X xzy � � { � - �r5 � ! � ��_Ptwt]u<odV TWV�Y*TxTWV=X ��V=\�haX q�[dYIZ=V o&\�R9TBYI_aR<t

K + 2K ′ + 2 = ∆ − 1��Y��}hrgDo&\(ikXl[]X�� t TBYI[WtS\�f t]_! lX

∆ + 1�HZ=hau<t�TWV=X�TW[]XlXyP u<tmTS`;XlR9TW_a\�R=X����EZ=hau<t�\�R=X

[]\F\�T�o&\�R=R=X�o TWX���TW\6\�R=X ikXl[mTWX&^�_aR X(Y�odV�\�f�TWV=X∆o&\�R=R=X�o TWX���o&\�`;ZH\�R=XlR9Tdt�[]X&ikX(YIhaX�� N6��\�R<t]X �9u=XlR9TWhrg��

TWV=X1�=R�YIh1q�[dYIZ=V�_Pt�Y�TW[]XlXw\�f�`�Y*^�_a`�u=`v�}Xlq�[]XlX∆YIR<�|\�f�\�[W�}Xl[

∆(∆ + 1) + 1g<t]\

N = O(log n√

n)N

"�V=X\�=R�YIh�ikXl[mTWX&^o&\(ikXl[;o&\�R<t TW[]u<o TWX�� �9g Z=h:YlgkXl[�y6V�Y�t�t]_! lX |C| > ∆2 � |S| 6 ∆ + 1�BN�Q�R�TWV=X

\�TWV=Xl[�V�YIR<���HY�`;_aR=_a`�u=` ikXl[mTWX&^�o&\(ikXl[�\�f�TWV=Xwq�[dYIZ=V6_Pt�\�f tm_! lX∆ + 2

YIR<��TWV=XwZ=[]\F\�f�\�f�TWV=Xl\�[]Xl` F�_Pto&\�`;Z=haX&TWX�N

y �