Hindawi Publishing CorporationISRN CombinatoricsVolume 2013 Article ID 795427 7 pageshttpdxdoiorg1011552013795427

Research ArticleDomination Integrity of Splitting Graph of Path and Cycle

Samir K Vaidya1 and Nirang J Kothari2

1 Department of Mathematics Saurashtra University Rajkot 360005 Gujarat India2 B H Gardi College of Engineering and Technology PO Box 215 Rajkot 360001 Gujarat India

Correspondence should be addressed to Samir K Vaidya samirkvaidyayahoocoin

Received 9 October 2012 Accepted 19 November 2012

Academic Editors C da Fonseca R Dondi and R Sotirov

Copyright copy 2013 S K Vaidya and N J Kothari is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

If 119878119878 is a dominating set of a connected graph 119866119866 then the domination integrity is the minimum of the sum of two parameters thenumber of elements in 119878119878 and the order of the maximum component of119866119866119866119878119878 We investigate domination integrity of splitting graphof path 119875119875119899119899 and cycle 119862119862119899119899 is work is an effort to relate network expansion and vulnerability parameter

1 Introduction

e stability of a communication network is of primeimportance for any network designer A shrinking networkeventually loses links or nodes its effectiveness is contin-uously decreasing and network becomes vulnerable Manygraph theoretic parameters have been introduced for themeasurement of vulnerability Some of them are connectivitytoughness integrity and rupture degree e integrity ofa graph is one of the well explored concepts which wasintroduced by Barefoot et al [1]

Denition 1 e integrity of a graph 119866119866 is denoted by 119868119868119868119866119866119868and dened by 119868119868119868119866119866119868 =min|119878119878|119878119878119878119868119866119866119866119878119878119868 119878 119878119878 119878 119878119878119868119866119866119868119878 where119878119878119868119866119866 119866 119878119878119868 is the order of a maximum component of 119866119866 119866 119878119878

Denition 2 An 119868119868-set of 119866119866 is any (proper) subset 119878119878 of 119878119878119868119866119866119868for which 119868119868119868119866119866119868 119868 |119878119878| 119878 119878119878119868119866119866 119866 119878119878119868

e integrity of the complete graph119870119870119899119899 path119875119875119899119899 cycle119862119862119899119899star 1198701198701119899119899 complete bipartite graph 119870119870119878119878119899119899 and power graph ofcycle were discussed by Barefoot et al [1 2] while Goddardand Swart [3 4] have investigated the bounds for integrity ofgraphs and its complement ey have also investigated theintegrity of graphs in the context of some graph operationse integrity of middle graphs is discussed by Mamut and

Vumar [5] while integrity of total graphs is discussed byDuumlndar and Aytaccedil [6]

Denition 3 A set 119878119878 119878 119878119878 of vertices in a graph 119866119866 119868 119868119878119878 119866119866119868 iscalled dominating set if every vertex 119907119907 119907 119878119878 is either an elementof 119878119878 or is adjacent to an element of 119878119878

Denition e domination number 120574120574119868119866119866119868 of a graph 119866119866equals the minimum cardinality of minimal dominating setof graph 119866119866

If 119863119863 is any minimal dominating set and if the order ofthe largest component of 119866119866 119866119863119863 is small then the removal of119863119863 will crash the communication network Considering thisaspect the concept of domination integrity was introducedby Sundareswaran and Swaminathan [7]

Denition 5 e domination integrity of a connected graph119866119866 is denoted as119863119863119868119868119868119866119866119868=min|119878119878|119878119878119878119868119866119866119866119878119878119868119878119878119878 is a dominatingset119878 where 119878119878119868119866119866 119866 119878119878119868 is the order of a maximum componentof 119866119866 119866 119878119878

Sundareswaran and Swaminathan [8] have investigateddomination integrity of middle graph of some graphs whileVaidya and Kothari [9] have discussed domination integrityof a graph obtained by duplication of an edge by a vertex and

2 ISRN Combinatorics

duplication of vertex by an edge in 119862119862119899119899 and 119875119875119899119899 In the presentwork we investigate domination integrity of splitting graphsof path and cycle

enition For a graph119866119866 the splitting graph 119878119878prime(119866119866119866 of graph119866119866 is obtained by adding a new vertex 119907119907prime corresponding to eachvertex 119907119907 of 119866119866 such that119873119873(119907119907119866 119873 119873119873(119907119907prime119866 where119873119873(119907119907119866 and119873119873(119907119907prime119866are the neighborhood sets of 119907119907 and 119907119907prime respectively

For standard graph theoretic notation we refer to West[10] while for terminology related to domination in Graphswe refer to Haynes et al [11]

2 Main Results

eorem 7 For all 119899119899 119899 119899

120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 119873

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119899119899119899

if 119899119899 119899 119899 (mod 4119866

119899119899 119899 119899119899

if 119899119899 119899 119899119899 119899 (mod 4119866

119899119899 119899 119899119899

if 119899119899 119899 119899 (mod 4119866

(1)

Proof Let 119907119907119899119899 119907119907119899119899hellip 119899 119907119907119899119899 be the vertices of path 119875119875119899119899 and119906119906119899119899 119906119906119899119899hellip 119899 119906119906119899119899 be the vertices corresponding to 119907119907119899119899 119907119907119899119899hellip 119899 119907119907119899119899which are added to obtain 119878119878prime(119875119875119899119899119866 As 119873119873(119906119906119899119866 119873 119907119907119899 and119873119873(119906119906119899119899119866 119873 119907119907119899119899119899119899 at least one vertex from each pair from 119906119906119899119899 119907119907119899and 119906119906119899119899119899 119907119907119899119899119899119899 must belong to any dominating set of 119878119878prime(119875119875119899119899119866Also at least one vertex from 119907119907119894119894119899119899 and 119907119907119894119894119899119899 must belong toany dominating set as 119873119873(119906119906119894119894119866 119873 119907119907119894119894119899119899119899 119907119907119894119894119899119899 and 119873119873(119907119907119894119894119866 119873119907119907119894119894119899119899119899 119907119907119894119894119899119899119899 119906119906119894119894119899119899119899 119906119906119894119894119899119899 where 119899 le 119894119894 le 119899119899 119899 119899 Consequently

|119878119878| 119899119899119899119899

for any dominating set 119878119878 (2)

Now depending upon the number of vertices in 119875119875119899119899 weconsider following subsets for 119899 le 119895119895 119895 1198951198991198991198954119895

119878119878 119873 100768210076821199071199071198991198994119895119895119899 119907119907119899119899411989511989510076981007698 119899 |119878119878| 119873119899119899119899

for 119899119899 119899 119899 (mod 4119866 119899

119878119878 119873 100768210076821199071199071198991198994119895119895119899 1199071199071198991198994119895119895119899 11990711990711989911989911989911989910076981007698 119899 |119878119878| 119873119899119899 119899 119899119899

for 119899119899 119899 119899 (mod 4119866 119899

119878119878 119873 100768210076821199071199071198991198994119895119895119899 1199071199071198991198994119895119895119899 119907119907119899119899119899119899119899 11990711990711989911989911989911989910076981007698 119899 |119878119878| 119873119899119899 119899 119899119899

for 119899119899 119899 119899 (mod 4119866 119899

119878119878 119873 100768210076821199071199071198991198994119895119895119899 1199071199071198991198994119895119895119899 119907119907119899119899119899119899119899 11990711990711989911989911989911989910076981007698 119899 |119878119878| 119873119899119899 119899 119899119899

for 119899119899 119899 119899 (mod 4119866

(3)

We claim that each 119878119878 is dominating set because119873119873(1199071199071198991198994119895119895119866 119873 1199071199071198991198994119895119895119899 1199071199071198991198994119895119895119899 1199061199061198991198994119895119895119899 1199061199061198991198994119895119895 and 119873119873(1199071199071198991198994119895119895119866 1198731199071199071198991198994119895119895119899 11990711990741198994119895119895119899 1199061199061198991198994119895119895119899 11990611990641198994119895119895

Also each 119878119878 is minimal because the vertex 1199061199061198991198994119895119895 will notbe dominated by any of the vertices when the vertex 1199071199071198991198994119895119895 isremoved

us above dened 119878119878 is minimal dominating set of119878119878prime(119875119875119899119899119866 hence from (2) we get

120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 119873

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119899119899119899

if 119899119899 119899 119899 (mod 4119866

119899119899 119899 119899119899

if 119899119899 119899 119899119899 119899 (mod 4119866

119899119899 119899 119899119899

if 119899119899 119899 119899 (mod 4119866

(4)

Observation 1 If 119878119878 is any dominating set of 119878119878prime(119875119875119899119899119866 with119898119898(119878119878prime(119875119875119899119899119866 119899 119878119878119866 119873 119899 then |119878119878| 119899 119899119899 119899 119899

Observation 2 If 119878119878 is any dominating set of 119878119878prime(119875119875119899119899119866 with119898119898(119878119878prime(119875119875119899119899119866 119899 119878119878119866 119873 119899 then |119878119878| 119899 119899119899

eorem 8

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 119873

10076181007618100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076261007626100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119899 if 119899119899 119873 119899119899119899 if 119899 ⩽ 119899119899 ⩽ 119899119899119899 119899 119899 if 119899119899 ⩽ 119899119899 ⩽ 119899119899119899119899 119899 119899 if 119899119899 ⩽ 119899119899 ⩽ 119899119899119899119899 119899 119899 if 1198996 ⩽ 119899119899 ⩽ 119899119899119899119899 if 119899119899 119873 119899119899

(5)

Proof Let 119907119907119899119899 119907119907119899119899hellip 119899 119907119907119899119899 be the vertices of path 119875119875119899119899 and119906119906119899119899 119906119906119899119899hellip 119899 119906119906119899119899 be the vertices corresponding to 119907119907119899119899 119907119907119899119899hellip 119899 119907119907119899119899which are added to obtain 119878119878prime(119875119875119899119899119866

Case 1 (119899119899 119873 119899119866 From eorem 7 120574120574(119878119878prime(119875119875119899119866119866 119873 119899 and 119863119863 119873119907119907119899119899 119907119907119899 is a 120574120574-set of 119878119878

prime(119875119875119899119866 en 119898119898(119878119878prime(119875119875119899119866 119899 119863119863119866 119873 119899 isimplies that119863119863119863119863(119878119878prime(119875119875119899119866119866 119873 120574120574(119878119878

prime(119875119875119899119866119866119899119898119898(119878119878prime(119875119875119899119866119899119863119863119866 119873 119899119899119899 119873

119899 since 120574120574(119878119878prime(119875119875119899119866119866 le |119878119878| and 119898119898(119878119878prime(119875119875119899119866 119899 119863119863119866 le 119898119898(119878119878prime(119875119875119899119866 119899 119878119878119866for any dominating set 119878119878 of 119878119878prime(119875119875119899119866 Consequently 120574120574(119878119878

prime(119875119875119899119866119866 119899119898119898(119878119878prime(119875119875119899119866 119899119863119863119866 le |119878119878| 119899119898119898(119878119878

prime(119875119875119899119866 119899 119878119878119866 for any dominating set 119878119878of 119878119878prime(119875119875119899119866 Hence119863119863119863119863(119878119878

prime(119875119875119899119866119866 119873 119899

Case 2 (119899 le 119899119899 le 119899119866 Consider the following subcases

Subcase (i) (119899119899 119873 119899119866 Fromeorem 7 120574120574(119878119878prime(119875119875119899119866119866 119873 119899 and 119863119863 119873119907119907119899119899 119906119906119899 is a 120574120574-set of 119878119878

prime(119875119875119899119866 en 119898119898(119878119878prime(119875119875119899119866 119899 119863119863119866 119873 119899 isimplies that119863119863119863119863(119878119878prime(119875119875119899119866119866 119873 120574120574(119878119878

prime(119875119875119899119866119866119899119898119898(119878119878prime(119875119875119899119866119899119863119863119866 119873 119899119899119899 119873

119899 since 120574120574(119878119878prime(119875119875119899119866119866 le |119878119878| and 119898119898(119878119878prime(119875119875119899119866 119899 119863119863119866 le 119898119898(119878119878prime(119875119875119899119866 119899 119878119878119866for any dominating set 119878119878 of 119878119878prime(119875119875119899119866 Consequently 120574120574(119878119878

prime(119875119875119899119866119866 119899119898119898(119878119878prime(119875119875119899119866 119899119863119863119866 le |119878119878| 119899119898119898(119878119878

prime(119875119875119899119866 119899 119878119878119866 for any dominating set 119878119878of 119878119878prime(119875119875119899119866 Hence119863119863119863119863(119878119878

prime(119875119875119899119866119866 119873 119899

Subcase (ii) (4 le 119899119899 le 6119866 From eorem 7 120574120574(119878119878prime(119875119875119899119899119866119866 119873 119899119899 119899 119899and119863119863 119873 119907119907119899119899 119907119907119899119899hellip 119899 119907119907119899119899119899119899 is a 120574120574-set of 119878119878

prime(119875119875119899119899119866en119898119898(119878119878prime(119875119875119899119899119866119899119863119863119866 119873 119899

ISRN Combinatorics 3

erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11986311986310076651007665

= 119899119899 minus 119899 + 119899 = 119899119899119899(6)

If 119878119878 is any dominating set other than 119863119863 of 119878119878prime(119875119875119899119899) with119898119898(119878119878prime(119875119875119899119899) minus 119878119878) = 119878 then |119878119878| 119878 119899119899 is implies that

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987810076651007665 119878 119899119899 + 119878 119899 119899119899119899 (7)

Hence from (6) and (7)119863119863119863119863(119878119878prime(119875119875119899119899)) = 119899119899

Subcase (iii) (119899119899 = 119899) Fromeorem 7 120574120574(119878119878prime(119875119875119899)) = 4 and119863119863 =119907119907119899 1199071199073 1199071199075 1199071199076 is a 120574120574-set of 119878119878

prime(119875119875119899) en119898119898(119878119878prime(119875119875119899) minus 119863119863) = 3

erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989910076651007665 minus 11986311986310076651007665 = 4 + 3 = 119899119899

(8)

Now if 119878119878119878 is any dominating set other than 119863119863 of 119878119878prime(119875119875119899) and119898119898(119878119878prime(119875119875119899) minus 119878119878119878) = 119899 then |119878119878119878| 119878 5 is implies that

1009249100924911987811987811987810092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989910076651007665 minus 11987811987811987810076651007665 119878 5 + 119899 = 119899119899 (9)

Also if 119878119878119899 is any dominating set other than119863119863 and 119878119878119878 of 119878119878prime(119875119875119899)

with119898119898(119878119878prime(119875119875119899) minus 119878119878119899) = 119878 then |119878119878119899| 119878 119899is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989910076651007665 minus 11987811987811989910076651007665 119878 119899 + 119878 = 8119899 (10)

Hence from (8) (9) and (10)119863119863119863119863(119878119878prime(119875119875119899)) = 119899

Case 3 (119899119899 = 119878119899) From eorem 7 120574120574(119878119878prime(119875119875119878119899)) = 6 and 119863119863 =119907119907119899 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 is a 120574120574-set of 119878119878

prime(119875119875119878119899) en 119898119898(119878119878prime(119875119875119878119899) minus119863119863) = 3

erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198781198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198781198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511987811989910076651007665 minus 11986311986310076651007665

= 6 + 3 = 9119899(11)

Now if 119878119878119878 is any dominating set other than 119863119863 of 119878119878prime(119875119875119878119899) and119898119898(119878119878prime(119875119875119878119899) minus 119878119878119878) = 119899 then |119878119878119878| 119878 8

is implies that

1009249100924911987811987811987810092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511987811989910076651007665 minus 11987811987811987810076651007665 119878 8 + 119899 = 119878119899119899 (12)

Also if 119878119878119899 is any dominating set other than119863119863 and 119878119878119878 of 119878119878prime(119875119875119878119899)

with119898119898(119878119878prime(119875119875119878119899) minus 119878119878119899) = 119878 then |119878119878119899| 119878 119878119899is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511987811989910076651007665 minus 11987811987811989910076651007665 119878 119878119899 + 119878 = 119878119878119899 (13)

Hence from (11) (12) and (13)119863119863119863119863(119878119878prime(119875119875119878119899)) = 9

Case 4 (119899119899 = 8 9 and 119878119878 le 119899119899 le 1198789) We know that

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11986311986310076651007665 119899 (14)

e domination number with 120574120574(119878119878prime(119862119862119899119899)) +119898119898(119878119878prime(119875119875119899119899) minus119863119863) and

set 119878119878 with |119878119878| + 119898119898(119878119878prime(119875119875119899119899) minus 119878119878) are shown in Table 1From Table 1

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987810076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11986311986310076651007665 119899

(15)

If 119878119878119878 is any dominating set of 119878119878prime(119875119875119899119899) other than119863119863 and 119878119878 with119898119898(119878119878prime(119875119875119899119899) minus 119878119878119878) = 4 or 5 then |119878119878| le |119878119878119878|

is implies that

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987810076651007665 le 1009249100924911987811987811987810092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987811987810076651007665 119899 (16)

If 119878119878119899 is any dominating set other than 119863119863 119878119878119878 and 119878119878 with119898119898(119878119878prime(119875119875119899119899) minus 119878119878119899) = 119899 then |119878119878119899| 119878 119899119899 minus 119899

is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987811989910076651007665 119878 (119899119899 minus 119899) + 119899 = 119899119899119899 (17)

If 1198781198783 is any dominating set other than 119863119863 119878119878119878 119878119878119899 and 119878119878 with119898119898(119878119878prime(119875119875119899119899) minus 1198781198783) = 119878 then |1198781198783| 119878 119899119899

is implies that

10092491009249119878119878310092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 119878119878310076651007665 119878 119899119899 + 119878119899 (18)

Hence from Table 1 and the results (14) to (18) we concludethat

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 = |119878119878| + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987810076651007665

=

10076181007618100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076261007626100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

3 if 119899119899 = 119899119899119899 if 3 ⩽ 119899119899 ⩽ 9119899119899 minus 119878 if 119878119899 ⩽ 119899119899 ⩽ 119878119899119899119899 minus 119899 if 1198783 ⩽ 119899119899 ⩽ 1198785119899119899 minus 3 if 1198786 ⩽ 119899119899 ⩽ 11987881198785 if 119899119899 = 1198789119899

(19)

Hence the result

eorem 9

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 = 120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 + 6 forall119899119899 119878 119899119899119899 (20)

Proof Let 119907119907119878 119907119907119899hellip 119907119907119899119899 be the vertices of path 119875119875119899119899 and119906119906119878 119906119906119899hellip 119906119906119899119899 be the vertices corresponding to 119907119907119878 119907119907119899hellip 119907119907119899119899which are added to obtain 119878119878prime(119875119875119899119899) en fromeorem 7

120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 =

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119899119899119899

if 119899119899 119899 119899 (mod 4)

119899119899 + 119878119899

if 119899119899 119899 119878 3 (mod 4)

119899119899 + 119899119899

if 119899119899 119899 119899 (mod 4)

(21)

4 ISRN Combinatorics

T 1

119899119899 119899119899119899119899119899prime119899119875119875119899119899)) 119898119898119899119899119899prime119899119875119875119899119899) minus 119863119863) 119899119899119899119899119899prime119899119875119875119899119899)) + 119898119898119899119899119899prime119899119875119875119899119899) minus 119863119863) 119899119899 119898119898119899119899119899prime119899119875119875119899119899) minus 119899119899) 119878119899119899119878 + 119898119898119899119899119899prime119899119875119875119899119899) minus 119899119899)

8 4 6 10 1199071199072 1199071199073 1199071199074 1199071199076 1199071199077 3 89 5 6 11 1199071199072 1199071199073 1199071199075 1199071199076 1199071199077 1199071199078 3 911 6 6 12 1199071199072 1199071199073 1199071199075 1199071199076 1199071199077 1199071199079 11990711990710 3 1012 6 6 12 1199071199072 1199071199073 1199071199075 1199071199076 1199071199077 1199071199078 11990711990710 11990711990711 3 1113 7 6 13 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 3 1114 8 6 14 1199071199072 1199071199073 1199071199074 1199071199076 1199071199077 1199071199079 11990711990710 11990711990712 11990711990713 3 1215 8 6 14 1199071199072 1199071199073 1199071199074 1199071199076 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1316 8 6 14 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 3 1317 9 6 15 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 11990711990716 3 1418 10 6 16 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 11990711990716 11990711990717 3 1519 10 6 16 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 11990711990717 11990711990718 3 15

119863119863 119863 1199071199074119895119895+2 1199071199074119895119895+3 where 0 le 119895119895 119895 1198951198991198991198954119895 for 119899119899 119899 0 119899119899119899119899 4)119863119863 119863 1199071199074119895119895+2 1199071199074119895119895+3 119907119907119899119899minus1 where 0 le 119895119895 119895 1198951198991198991198954119895 minus 1 for 119899119899 1198991 119899119899119899119899 4) 119863119863 119863 1199071199074119895119895+2 1199071199074119895119895+3 119907119907119899119899minus2 119907119907119899119899minus1 where 0 le 119895119895 119895 1198951198991198991198954119895for 119899119899 119899 2 3 1198991198991198991198994) are 119899119899-sets of 119899119899prime119899119875119875119899119899) en 119898119898119899119899119899prime119899119875119875119899119899) minus119863119863) 119863 6

erefore

11986311986311986311986310076511007651119899119899prime 100764910076491198751198751198991198991007665100766510076651007665 le 119899119899 10076511007651119899119899prime 100764910076491198751198751198991198991007665100766510076651007665 + 6 (22)

Now we claim that if 119898119898119899119898119898 minus 119899119899) 119898 6 for any dominating set 119899119899other than119863119863 then

119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911987511987511989911989910076651007665 minus 11989911989910076651007665 ge 119899119899 10076511007651119899119899prime 100764910076491198751198751198991198991007665100766510076651007665 + 6 (23)

Case 1 If 1198991198991 is any dominating set other than 119863119863 and119898119898119899119899119899prime119899119875119875119899119899)minus1198991198991) gt 4 or 5 then 1198781198991198991119878 ge 119899119899119899119899119899

prime119899119875119875119899119899))+2 Consequently1198781198991198991119878 + 119898119898119899119899119899

prime119899119875119875119899119899) minus 1198991198991) ge 119899119899119899119899119899prime119899119875119875119899119899)) + 6

Case 2 If 1198991198992 is any dominating set other than 119863119863 and 1198991198991 of119899119899prime119899119875119875119899119899) with 119898119898119899119899119899prime119899119875119875119899119899) minus 1198991198992) 119863 3 then 1198781198991198992119878 ge 119899119899119899 minus 1198991198991198991198953119899)Consequently 1198781198991198992119878 + 119898119898119899119899119899prime119899119875119875119899119899) minus 1198991198992) ge 119899119899119899 minus 1198991198991198991198953119899) + 3 gt1198991198991198991198991198952119899 + 1) + 6 119863 119899119899119899119899119899prime119899119875119875119899119899)) + 6

Case 3 If 1198991198993 is any dominating set other than119863119863 1198991198991 and 1198991198992 of119899119899prime119899119875119875119899119899)with119898119898119899119899119899

prime119899119875119875119899119899)minus1198991198993) 119863 2 then 1198781198991198993119878 ge 119899119899minus2 Consequently1198781198991198993119878 + 119898119898119899119899119899

prime119899119875119875119899119899) minus 1198991198993) ge 119899119899119899 minus 2) + 2 119863 119899119899 gt 1198991198991198991198991198952119899 + 1) + 6 119863119899119899119899119899119899prime119899119875119875119899119899)) + 6

Case 4 If 1198991198994 is any dominating set other than119863119863 1198991198991 1198991198992 and 1198991198993of 119899119899prime119899119875119875119899119899)with119898119898119899119899119899

prime119899119875119875119899119899)minus1198991198994) 119863 1 then 1198781198991198994119878 ge 119899119899 Consequently1198781198991198994119878 +119898119898119899119899119899

prime119899119875119875119899119899)minus1198991198994) ge 119899119899+1 gt 1198991198991198991198991198952119899+1)+6 119863 119899119899119899119899119899prime119899119875119875119899119899))+6

us from (22) and (23)we have119863119863119863119863119899119899119899prime119899119875119875119899119899)) 119863 119899119899119899119899119899prime119899119875119875119899119899))+

6

eorem 10 For all 119899119899 ge 3

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076651007665 119863

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899119899119899119899 4)

119899119899 + 12

if 119899119899 119899 1 3 119899119899119899119899 4)

119899119899 + 22

if 119899119899 119899 2 119899119899119899119899 4)

(24)

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119899119899prime119899119862119862119899119899) Now either of the vertexfrom 119907119907119894119894minus1 and 119907119907119894119894+1 must belong to any dominating set 119899119899 as119873119873119899119906119906119894119894) 119863 119907119907119894119894minus1 119907119907119894119894+1

Consequently

119878119899119899119878 ge1198991198992for any dominating set 119899119899 (25)

Now depending upon the different possibilities of 119899119899 wechoose 119899119899 as follows

For 119899119899 119899 0 119899119899119899119899 4) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 with 119878119899119899119878 1198631198991198991198952 (where 1 le 119895119895 119895 1198951198991198991198954119895)for 119899119899 119899 1 119899119899119899119899 4) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 with 119878119899119899119878 119863 119899119899119899 +1)1198952 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 2 1198991198991198991198994) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 with 119878119899119899119878 119863119899119899119899 + 2)1198952 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 3 1198991198991198991198994) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 with 119878119899119899119878 119863119899119899119899 + 1)1198952 (where 1 le 119895119895 le 1198951198991198991198954119895)

We claim that each 119899119899 is a minimal dominating setof 119899119899prime119899119862119862119899119899) since 1198731198731198991199071199071) 119863 1199071199072 119907119907119899119899 1199061199062 119906119906119899119899 1198731198731198991199071199074119895119895) 1198631199071199074119895119895minus1 1199071199074119895119895+1 1199061199064119895119895minus1 1199061199064119895119895+1 1198731198731198991199071199074119895119895+1) 119863 1199071199074119895119895 1199071199074119895119895+2 1199061199064119895119895 1199061199064119895119895+2and 119873119873119899119907119907119899119899) 119863 1199071199071 119907119907119899119899minus1 1199061199061 119906119906119899119899minus1 and removal of 1199071199074119895119895 a vertex1199061199064119895119895+1 will not be dominated by any vertex hence from (25)

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076651007665 119863

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899119899119899119899 4)

119899119899 + 12

if 119899119899 119899 1 3 119899119899119899119899 4)

119899119899 + 22

if 119899119899 119899 2 119899119899119899119899 4)

(26)

eorem 11

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076651007665 119863100761810076181007626100762610076421007642

119899119899 + 1119899 3 le 119899119899 le 6

119899119899 minus 10078121007812119899119899310078281007828 + 3119899 7 le 119899119899 le 18

(27)

ISRN Combinatorics 5

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119878119878prime(119862119862119899119899)

Case 1 (119899119899 119899 119899) From eorem 10 120574120574(119878119878prime(119862119862119899)) 119899 2 and 119863119863 1198991199071199071 1199071199072 is a 120574120574-set of 119878119878

prime(119862119862119899) then119898119898(119878119878prime(119862119862119899) minus119863119863) 119899 119899 erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198621198621198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 11986311986310076651007665

119899 2 + 119899 119899 5(28)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(119862119862119899) with119898119898(119878119878prime(119862119862119899) minus 1198781198781) 119899 2 then |1198781198781| ge 119899 is implies that

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 119878119878110076651007665 ge 2 + 119899 119899 5 (29)

Let 1198781198782 119899 1199071199071 1199071199072 119907119907119899 be dominating set of 119878119878prime(119862119862119899) and119898119898(119878119878prime(119862119862119899) minus 1198781198782) 119899 1 is implies that

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 119878119878210076651007665 119899 119899 + 1 119899 4 (30)

Hence from (28) (29) and (30)119863119863119863119863(119878119878prime119862119862119899) 119899 4

Case 2 (119899119899 119899 4) From eorem 10 120574120574(119878119878prime(1198621198624)) 119899 2 and 119863119863 1198991199071199071 1199071199072 is a 120574120574-set of 119878119878

prime(1198621198624) then119898119898(119878119878prime(1198621198624) minus 119863119863) 119899 119899

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986241007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986241007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862410076651007665 minus 11986311986310076651007665

119899 2 + 119899 119899 5(31)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198624) and119898119898(119878119878prime(1198621198624) minus 1198781198781) 119899 2 or 1 then |1198781198781| ge 4 erefore

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862410076651007665 minus 119878119878110076651007665 ge 4 + 2 119899 6 (32)

Hence from (31) and (32)119863119863119863119863(119878119878prime(1198621198624)) 119899 5

Case 3 (119899119899 119899 5) From eorem 10 120574120574(119878119878prime(1198621198625)) 119899 119899 and 119863119863 1198991199071199072 119907119907119899 1199071199074 is a 120574120574-set of 119878119878

prime(1198621198625) then119898119898(119878119878prime(1198621198625) minus 119863119863) 119899 6

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 11986311986310076651007665

119899 119899 + 6 119899 9(33)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198625) and119898119898(119878119878prime(1198621198625) minus 1198781198781) 119899 4 or 5 then |1198781198781| ge 120574120574(119878119878

prime(1198621198625)) 119899 119899is implies

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 119878119878110076651007665 ge 120574120574 10076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 + 4

119899 119899 + 4 119899 7(34)

If 1198781198782 is any dominating set other than 119863119863 and 1198781198781 with119898119898(119878119878prime(1198621198625) minus 1198781198782) 119899 2 or 119899 then |1198781198782| ge 5 is implies that

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 119878119878210076651007665 ge 5 + 2 119899 7 (35)

Let 119878119878119899 119899 1199071199071 1199071199072 119907119907119899 1199071199074 1199071199075 be dominating set of 119878119878prime(1198621198625) then119898119898(119878119878prime(1198621198625) minus 119878119878119899) 119899 1 is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 11987811987811989910076651007665 119899 5 + 1 119899 6 (36)

Hence from (33) (34) (35) and (36)119863119863119863119863(119878119878prime(1198621198625)) 119899 6

Case 4 (119899119899 119899 6) From eorem 10 120574120574(119878119878prime(1198621198626)) 119899 4 and 119863119863 1198991199071199071 1199071199072 1199071199074 1199071199075 is a 120574120574-set of 119878119878

prime(1198621198626) then119898119898(119878119878prime(1198621198626) minus 119863119863) 119899 119899

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986261007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986261007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 11986311986310076651007665

119899 4 + 119899 119899 7(37)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198626) and119898119898(119878119878prime(1198621198626) minus 1198781198781) 119899 2 then |1198781198781| ge 6 is implies that

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 119878119878110076651007665 ge 6 + 2 119899 8 (38)

Let 1198781198782 119899 1199071199071 1199071199072 119907119907119899 1199071199074 1199071199075 1199071199076 be dominating set of 119878119878prime(1198621198626)then119898119898(119878119878prime(1198621198626) minus 1198781198782) 119899 1

erefore

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 119878119878210076651007665 119899 6 + 1 119899 7 (39)

Hence from (37) (38) and (39)119863119863119863119863(119878119878prime(1198621198626)) 119899 7

Case 5 (7 le 119899119899 le 18) We know that

11986311986311986311986310076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11986311986310076651007665 (40)

e domination number with 120574120574(119878119878prime(119862119862119899119899)) +119898119898(119878119878prime(119862119862119899119899) minus119863119863) and

set 119878119878 with |119878119878| + 119898119898(119878119878prime(119862119862119899119899) minus 119878119878) are shown in Table 2From Table 2 for any dominating set 119878119878 other than 119863119863 of

119878119878prime(119862119862119899119899) we have

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11987811987810076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11986311986310076651007665

(41)

If 1198781198781 is any dominating set of 119878119878prime(119862119862119899119899) other than119863119863 and 119878119878 with119898119898(119878119878prime(119862119862119899119899) minus 1198781198781) 119899 4 or 5 then |119878119878| le |1198781198781|

is implies that

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11987811987810076651007665 le 10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 119878119878110076651007665 (42)

If 1198781198782 is any dominating set of 119878119878prime(119862119862119899119899) other than 119863119863 119878119878 and 1198781198781with119898119898(119878119878prime(119862119862119899119899) minus 1198781198782) 119899 2 or 1 then |1198781198782| ge 119899119899

6 ISRN Combinatorics

T 2

119899119899 119899119899119899119899119899prime119899119862119862119899119899)) 119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) 119899119899119899119899119899prime119899119862119862119899119899)) + 119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) 119899119899 119898119898119899119899119899prime119899119862119862119899119899) minus 119899119899) 119878119899119899119878 + 119898119898119899119899119899prime119899119862119862119899119899) minus 119899119899)

7 4 6 10 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 3 88 4 6 10 1199071199071 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 3 99 5 6 11 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 3 910 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 3 1011 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 3 1112 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990712 3 1113 7 6 13 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 3 1214 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1315 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1316 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 11990711990716 3 14

is implies that

10092491009249119899119899210092491009249 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 119899119899210076671007667 ge 119899119899 + 1119899 (43)

Hence from Table 2 and the results (40) to (43) we concludethat

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 = 119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 11989911989910076671007667

= 119899119899 minus 10078121007812119899119899310078281007828 + 3119899

(44)

Hence the result

eorem 12 For all 119899119899 ge 17 119863119863119863119863119899119899119899prime119899119862119862119899119899)) = 119899119899119899119899119899prime119862119862119899119899) + 6

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119899119899prime119899119862119862119899119899) en fromeorem 10

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 =

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899mod 4)

119899119899 + 12

if 119899119899 119899 1 3 119899mod 4)

119899119899 + 22

if 119899119899 119899 2 119899mod 4)

(45)

119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 (where 1 le 119895119895 119895 1198951198991198991198954119895) for 119899119899 1198990 119899mod4) 119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 (where 1 le 119895119895 le 1198951198991198991198954119895) for119899119899 119899 1 119899mod 4) 119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 2 119899mod4) or 119899119899 119899 3 119899mod4) are 119899119899-sets of 119899119899prime119899119862119862119899119899)en119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) = 6

erefore

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 le 119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 + 6119899 (46)

Now we claim that if 119898119898119899119899119899prime119899119862119862119899119899)) ne 6 for any dominating set 119899119899other than119863119863 then

119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 11989911989910076671007667 ge 119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 + 6119899 (47)

Case 1 If 1198991198991 is any dominating set other than 119863119863 and119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198991) = 4 or 5 then 1198781198991198991119878 ge 119899119899119899119899119899prime119899119862119862119899119899)) + 2Consequently 1198781198991198991119878 + 119898119898119899119899119899

prime119899119862119862119899119899) minus 1198991198991) ge 119899119899119899119899119899prime119899119862119862119899119899)) + 6

Case 2 If 1198991198992 is any dominating set other than 119863119863 and 1198991198991 of119899119899prime119899119862119862119899119899) with 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198992) = 3 then 1198781198991198992119878 ge 119899119899119899 minus 1198951198991198991198953119895)Consequently 1198781198991198992119878 + 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198992) ge 119899119899119899 minus 1198951198991198991198953119895) + 3 119899119899119899119899119899119899prime119899119862119862119899119899)) + 6

Case 3 If 1198991198993 is any dominating set other than 119863119863 1198991198991 and1198991198992 of 119899119899prime119899119862119862119899119899) with 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198993) = 2 or 1 then 1198781198991198993119878 ge 119899119899Consequently 1198781198991198993119878 + 119898119898119899119899119899

prime119899119862119862119899119899) minus 1198991198993) ge 119899119899 + 1 119899 119899119899119899119899119899prime119899119862119862119899119899)) + 6

us from (46) and (47) we have119863119863119863119863119899119899119899prime119899119862119862119899119899))

3 Concluding Remarks

e domination and vulnerability of network are two impor-tant concepts for the network security We have studied animportant measure of vulnerability known as dominationintegrity and investigate domination integrity of splittinggraphs of path and cycle e results reported here throwsome light in the direction to nd the domination integrityof larger graph obtained from the given graph

References

[1] C A Barefoot R Entringer and H C Swart ldquoVulnerabilityin Graphs-a Comparative Surveyrdquo Journal of CombinatorialMathematics and Combinatorial Computing vol 1 pp 13ndash221987

[2] C A Barefoot R Entringer and H C Swart ldquoIntegrity of treesand powers of cyclesrdquo Congressus Numerantium vol 58 pp103ndash114 1987

[3] W Goddard andH C Swart ldquoOn the integrity of combinationsof graphsrdquo Journal of CombinatorialMathematics andCombina-torial Computing vol 4 pp 3ndash18 1988

[4] W Goddard and H C Swart ldquoIntegrity in graphs bounds andbasicsrdquo Journal of Combinatorial Mathematics and Combinato-rial Computing vol 7 pp 139ndash151 1990

[5] A Mamut and E Vumar A Note on the Integrity of MiddleGraphs Lecture Notes in Computer Science vol 4381 SpringerNew York NY USA 2007

ISRN Combinatorics 7

[6] P Duumlndar and A Aytaccedil ldquoIntegrity of total graphs via certainparametersrdquoMathematical Notes vol 76 no 5-6 pp 665ndash6722004

[7] R Sundareswaran and V Swaminathan ldquoDomination Integrityin graphsrdquo in Proceedings of International conference on Mathe-matical and Experimental Physics pp 46ndash57 Narosa PublishingHouse 2010

[8] R Sundareswaran and V Swaminathan ldquoDomination integrityof middle graphsrdquo in Algebra Graph eory and eir Applica-tions T T Chelvam S Somasundaram and R Kala Eds pp88ndash92 Narosa Publishing House Delhi India 2010

[9] S Vaidya and N Kothari ldquoSome new results on dominationintegrity of graphsrdquo Open Journal of Discrete Mathematics vol2 pp 96ndash98 2012

[10] D B West Introduction To Graph eory 2e Prentice-HallNew Delhi India 2003

[11] T Haynes S Hedetniemi and P Slater Fundamentals ofDomination in Graphs Marcel Dekker New York NY USA1998

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Stochastic AnalysisInternational Journal of

2 ISRN Combinatorics

duplication of vertex by an edge in 119862119862119899119899 and 119875119875119899119899 In the presentwork we investigate domination integrity of splitting graphsof path and cycle

enition For a graph119866119866 the splitting graph 119878119878prime(119866119866119866 of graph119866119866 is obtained by adding a new vertex 119907119907prime corresponding to eachvertex 119907119907 of 119866119866 such that119873119873(119907119907119866 119873 119873119873(119907119907prime119866 where119873119873(119907119907119866 and119873119873(119907119907prime119866are the neighborhood sets of 119907119907 and 119907119907prime respectively

For standard graph theoretic notation we refer to West[10] while for terminology related to domination in Graphswe refer to Haynes et al [11]

2 Main Results

eorem 7 For all 119899119899 119899 119899

120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 119873

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119899119899119899

if 119899119899 119899 119899 (mod 4119866

119899119899 119899 119899119899

if 119899119899 119899 119899119899 119899 (mod 4119866

119899119899 119899 119899119899

if 119899119899 119899 119899 (mod 4119866

(1)

Proof Let 119907119907119899119899 119907119907119899119899hellip 119899 119907119907119899119899 be the vertices of path 119875119875119899119899 and119906119906119899119899 119906119906119899119899hellip 119899 119906119906119899119899 be the vertices corresponding to 119907119907119899119899 119907119907119899119899hellip 119899 119907119907119899119899which are added to obtain 119878119878prime(119875119875119899119899119866 As 119873119873(119906119906119899119866 119873 119907119907119899 and119873119873(119906119906119899119899119866 119873 119907119907119899119899119899119899 at least one vertex from each pair from 119906119906119899119899 119907119907119899and 119906119906119899119899119899 119907119907119899119899119899119899 must belong to any dominating set of 119878119878prime(119875119875119899119899119866Also at least one vertex from 119907119907119894119894119899119899 and 119907119907119894119894119899119899 must belong toany dominating set as 119873119873(119906119906119894119894119866 119873 119907119907119894119894119899119899119899 119907119907119894119894119899119899 and 119873119873(119907119907119894119894119866 119873119907119907119894119894119899119899119899 119907119907119894119894119899119899119899 119906119906119894119894119899119899119899 119906119906119894119894119899119899 where 119899 le 119894119894 le 119899119899 119899 119899 Consequently

|119878119878| 119899119899119899119899

for any dominating set 119878119878 (2)

Now depending upon the number of vertices in 119875119875119899119899 weconsider following subsets for 119899 le 119895119895 119895 1198951198991198991198954119895

119878119878 119873 100768210076821199071199071198991198994119895119895119899 119907119907119899119899411989511989510076981007698 119899 |119878119878| 119873119899119899119899

for 119899119899 119899 119899 (mod 4119866 119899

119878119878 119873 100768210076821199071199071198991198994119895119895119899 1199071199071198991198994119895119895119899 11990711990711989911989911989911989910076981007698 119899 |119878119878| 119873119899119899 119899 119899119899

for 119899119899 119899 119899 (mod 4119866 119899

119878119878 119873 100768210076821199071199071198991198994119895119895119899 1199071199071198991198994119895119895119899 119907119907119899119899119899119899119899 11990711990711989911989911989911989910076981007698 119899 |119878119878| 119873119899119899 119899 119899119899

for 119899119899 119899 119899 (mod 4119866 119899

119878119878 119873 100768210076821199071199071198991198994119895119895119899 1199071199071198991198994119895119895119899 119907119907119899119899119899119899119899 11990711990711989911989911989911989910076981007698 119899 |119878119878| 119873119899119899 119899 119899119899

for 119899119899 119899 119899 (mod 4119866

(3)

We claim that each 119878119878 is dominating set because119873119873(1199071199071198991198994119895119895119866 119873 1199071199071198991198994119895119895119899 1199071199071198991198994119895119895119899 1199061199061198991198994119895119895119899 1199061199061198991198994119895119895 and 119873119873(1199071199071198991198994119895119895119866 1198731199071199071198991198994119895119895119899 11990711990741198994119895119895119899 1199061199061198991198994119895119895119899 11990611990641198994119895119895

Also each 119878119878 is minimal because the vertex 1199061199061198991198994119895119895 will notbe dominated by any of the vertices when the vertex 1199071199071198991198994119895119895 isremoved

us above dened 119878119878 is minimal dominating set of119878119878prime(119875119875119899119899119866 hence from (2) we get

120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 119873

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119899119899119899

if 119899119899 119899 119899 (mod 4119866

119899119899 119899 119899119899

if 119899119899 119899 119899119899 119899 (mod 4119866

119899119899 119899 119899119899

if 119899119899 119899 119899 (mod 4119866

(4)

Observation 1 If 119878119878 is any dominating set of 119878119878prime(119875119875119899119899119866 with119898119898(119878119878prime(119875119875119899119899119866 119899 119878119878119866 119873 119899 then |119878119878| 119899 119899119899 119899 119899

Observation 2 If 119878119878 is any dominating set of 119878119878prime(119875119875119899119899119866 with119898119898(119878119878prime(119875119875119899119899119866 119899 119878119878119866 119873 119899 then |119878119878| 119899 119899119899

eorem 8

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 119873

10076181007618100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076261007626100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119899 if 119899119899 119873 119899119899119899 if 119899 ⩽ 119899119899 ⩽ 119899119899119899 119899 119899 if 119899119899 ⩽ 119899119899 ⩽ 119899119899119899119899 119899 119899 if 119899119899 ⩽ 119899119899 ⩽ 119899119899119899119899 119899 119899 if 1198996 ⩽ 119899119899 ⩽ 119899119899119899119899 if 119899119899 119873 119899119899

(5)

Proof Let 119907119907119899119899 119907119907119899119899hellip 119899 119907119907119899119899 be the vertices of path 119875119875119899119899 and119906119906119899119899 119906119906119899119899hellip 119899 119906119906119899119899 be the vertices corresponding to 119907119907119899119899 119907119907119899119899hellip 119899 119907119907119899119899which are added to obtain 119878119878prime(119875119875119899119899119866

Case 1 (119899119899 119873 119899119866 From eorem 7 120574120574(119878119878prime(119875119875119899119866119866 119873 119899 and 119863119863 119873119907119907119899119899 119907119907119899 is a 120574120574-set of 119878119878

prime(119875119875119899119866 en 119898119898(119878119878prime(119875119875119899119866 119899 119863119863119866 119873 119899 isimplies that119863119863119863119863(119878119878prime(119875119875119899119866119866 119873 120574120574(119878119878

prime(119875119875119899119866119866119899119898119898(119878119878prime(119875119875119899119866119899119863119863119866 119873 119899119899119899 119873

119899 since 120574120574(119878119878prime(119875119875119899119866119866 le |119878119878| and 119898119898(119878119878prime(119875119875119899119866 119899 119863119863119866 le 119898119898(119878119878prime(119875119875119899119866 119899 119878119878119866for any dominating set 119878119878 of 119878119878prime(119875119875119899119866 Consequently 120574120574(119878119878

prime(119875119875119899119866119866 119899119898119898(119878119878prime(119875119875119899119866 119899119863119863119866 le |119878119878| 119899119898119898(119878119878

prime(119875119875119899119866 119899 119878119878119866 for any dominating set 119878119878of 119878119878prime(119875119875119899119866 Hence119863119863119863119863(119878119878

prime(119875119875119899119866119866 119873 119899

Case 2 (119899 le 119899119899 le 119899119866 Consider the following subcases

Subcase (i) (119899119899 119873 119899119866 Fromeorem 7 120574120574(119878119878prime(119875119875119899119866119866 119873 119899 and 119863119863 119873119907119907119899119899 119906119906119899 is a 120574120574-set of 119878119878

prime(119875119875119899119866 en 119898119898(119878119878prime(119875119875119899119866 119899 119863119863119866 119873 119899 isimplies that119863119863119863119863(119878119878prime(119875119875119899119866119866 119873 120574120574(119878119878

prime(119875119875119899119866119866119899119898119898(119878119878prime(119875119875119899119866119899119863119863119866 119873 119899119899119899 119873

119899 since 120574120574(119878119878prime(119875119875119899119866119866 le |119878119878| and 119898119898(119878119878prime(119875119875119899119866 119899 119863119863119866 le 119898119898(119878119878prime(119875119875119899119866 119899 119878119878119866for any dominating set 119878119878 of 119878119878prime(119875119875119899119866 Consequently 120574120574(119878119878

prime(119875119875119899119866119866 119899119898119898(119878119878prime(119875119875119899119866 119899119863119863119866 le |119878119878| 119899119898119898(119878119878

prime(119875119875119899119866 119899 119878119878119866 for any dominating set 119878119878of 119878119878prime(119875119875119899119866 Hence119863119863119863119863(119878119878

prime(119875119875119899119866119866 119873 119899

Subcase (ii) (4 le 119899119899 le 6119866 From eorem 7 120574120574(119878119878prime(119875119875119899119899119866119866 119873 119899119899 119899 119899and119863119863 119873 119907119907119899119899 119907119907119899119899hellip 119899 119907119907119899119899119899119899 is a 120574120574-set of 119878119878

prime(119875119875119899119899119866en119898119898(119878119878prime(119875119875119899119899119866119899119863119863119866 119873 119899

ISRN Combinatorics 3

erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11986311986310076651007665

= 119899119899 minus 119899 + 119899 = 119899119899119899(6)

If 119878119878 is any dominating set other than 119863119863 of 119878119878prime(119875119875119899119899) with119898119898(119878119878prime(119875119875119899119899) minus 119878119878) = 119878 then |119878119878| 119878 119899119899 is implies that

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987810076651007665 119878 119899119899 + 119878 119899 119899119899119899 (7)

Hence from (6) and (7)119863119863119863119863(119878119878prime(119875119875119899119899)) = 119899119899

Subcase (iii) (119899119899 = 119899) Fromeorem 7 120574120574(119878119878prime(119875119875119899)) = 4 and119863119863 =119907119907119899 1199071199073 1199071199075 1199071199076 is a 120574120574-set of 119878119878

prime(119875119875119899) en119898119898(119878119878prime(119875119875119899) minus 119863119863) = 3

erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989910076651007665 minus 11986311986310076651007665 = 4 + 3 = 119899119899

(8)

Now if 119878119878119878 is any dominating set other than 119863119863 of 119878119878prime(119875119875119899) and119898119898(119878119878prime(119875119875119899) minus 119878119878119878) = 119899 then |119878119878119878| 119878 5 is implies that

1009249100924911987811987811987810092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989910076651007665 minus 11987811987811987810076651007665 119878 5 + 119899 = 119899119899 (9)

Also if 119878119878119899 is any dominating set other than119863119863 and 119878119878119878 of 119878119878prime(119875119875119899)

with119898119898(119878119878prime(119875119875119899) minus 119878119878119899) = 119878 then |119878119878119899| 119878 119899is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989910076651007665 minus 11987811987811989910076651007665 119878 119899 + 119878 = 8119899 (10)

Hence from (8) (9) and (10)119863119863119863119863(119878119878prime(119875119875119899)) = 119899

Case 3 (119899119899 = 119878119899) From eorem 7 120574120574(119878119878prime(119875119875119878119899)) = 6 and 119863119863 =119907119907119899 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 is a 120574120574-set of 119878119878

prime(119875119875119878119899) en 119898119898(119878119878prime(119875119875119878119899) minus119863119863) = 3

erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198781198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198781198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511987811989910076651007665 minus 11986311986310076651007665

= 6 + 3 = 9119899(11)

Now if 119878119878119878 is any dominating set other than 119863119863 of 119878119878prime(119875119875119878119899) and119898119898(119878119878prime(119875119875119878119899) minus 119878119878119878) = 119899 then |119878119878119878| 119878 8

is implies that

1009249100924911987811987811987810092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511987811989910076651007665 minus 11987811987811987810076651007665 119878 8 + 119899 = 119878119899119899 (12)

Also if 119878119878119899 is any dominating set other than119863119863 and 119878119878119878 of 119878119878prime(119875119875119878119899)

with119898119898(119878119878prime(119875119875119878119899) minus 119878119878119899) = 119878 then |119878119878119899| 119878 119878119899is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511987811989910076651007665 minus 11987811987811989910076651007665 119878 119878119899 + 119878 = 119878119878119899 (13)

Hence from (11) (12) and (13)119863119863119863119863(119878119878prime(119875119875119878119899)) = 9

Case 4 (119899119899 = 8 9 and 119878119878 le 119899119899 le 1198789) We know that

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11986311986310076651007665 119899 (14)

e domination number with 120574120574(119878119878prime(119862119862119899119899)) +119898119898(119878119878prime(119875119875119899119899) minus119863119863) and

set 119878119878 with |119878119878| + 119898119898(119878119878prime(119875119875119899119899) minus 119878119878) are shown in Table 1From Table 1

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987810076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11986311986310076651007665 119899

(15)

If 119878119878119878 is any dominating set of 119878119878prime(119875119875119899119899) other than119863119863 and 119878119878 with119898119898(119878119878prime(119875119875119899119899) minus 119878119878119878) = 4 or 5 then |119878119878| le |119878119878119878|

is implies that

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987810076651007665 le 1009249100924911987811987811987810092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987811987810076651007665 119899 (16)

If 119878119878119899 is any dominating set other than 119863119863 119878119878119878 and 119878119878 with119898119898(119878119878prime(119875119875119899119899) minus 119878119878119899) = 119899 then |119878119878119899| 119878 119899119899 minus 119899

is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987811989910076651007665 119878 (119899119899 minus 119899) + 119899 = 119899119899119899 (17)

If 1198781198783 is any dominating set other than 119863119863 119878119878119878 119878119878119899 and 119878119878 with119898119898(119878119878prime(119875119875119899119899) minus 1198781198783) = 119878 then |1198781198783| 119878 119899119899

is implies that

10092491009249119878119878310092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 119878119878310076651007665 119878 119899119899 + 119878119899 (18)

Hence from Table 1 and the results (14) to (18) we concludethat

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 = |119878119878| + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987810076651007665

=

10076181007618100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076261007626100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

3 if 119899119899 = 119899119899119899 if 3 ⩽ 119899119899 ⩽ 9119899119899 minus 119878 if 119878119899 ⩽ 119899119899 ⩽ 119878119899119899119899 minus 119899 if 1198783 ⩽ 119899119899 ⩽ 1198785119899119899 minus 3 if 1198786 ⩽ 119899119899 ⩽ 11987881198785 if 119899119899 = 1198789119899

(19)

Hence the result

eorem 9

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 = 120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 + 6 forall119899119899 119878 119899119899119899 (20)

Proof Let 119907119907119878 119907119907119899hellip 119907119907119899119899 be the vertices of path 119875119875119899119899 and119906119906119878 119906119906119899hellip 119906119906119899119899 be the vertices corresponding to 119907119907119878 119907119907119899hellip 119907119907119899119899which are added to obtain 119878119878prime(119875119875119899119899) en fromeorem 7

120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 =

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119899119899119899

if 119899119899 119899 119899 (mod 4)

119899119899 + 119878119899

if 119899119899 119899 119878 3 (mod 4)

119899119899 + 119899119899

if 119899119899 119899 119899 (mod 4)

(21)

4 ISRN Combinatorics

T 1

119899119899 119899119899119899119899119899prime119899119875119875119899119899)) 119898119898119899119899119899prime119899119875119875119899119899) minus 119863119863) 119899119899119899119899119899prime119899119875119875119899119899)) + 119898119898119899119899119899prime119899119875119875119899119899) minus 119863119863) 119899119899 119898119898119899119899119899prime119899119875119875119899119899) minus 119899119899) 119878119899119899119878 + 119898119898119899119899119899prime119899119875119875119899119899) minus 119899119899)

8 4 6 10 1199071199072 1199071199073 1199071199074 1199071199076 1199071199077 3 89 5 6 11 1199071199072 1199071199073 1199071199075 1199071199076 1199071199077 1199071199078 3 911 6 6 12 1199071199072 1199071199073 1199071199075 1199071199076 1199071199077 1199071199079 11990711990710 3 1012 6 6 12 1199071199072 1199071199073 1199071199075 1199071199076 1199071199077 1199071199078 11990711990710 11990711990711 3 1113 7 6 13 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 3 1114 8 6 14 1199071199072 1199071199073 1199071199074 1199071199076 1199071199077 1199071199079 11990711990710 11990711990712 11990711990713 3 1215 8 6 14 1199071199072 1199071199073 1199071199074 1199071199076 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1316 8 6 14 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 3 1317 9 6 15 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 11990711990716 3 1418 10 6 16 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 11990711990716 11990711990717 3 1519 10 6 16 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 11990711990717 11990711990718 3 15

119863119863 119863 1199071199074119895119895+2 1199071199074119895119895+3 where 0 le 119895119895 119895 1198951198991198991198954119895 for 119899119899 119899 0 119899119899119899119899 4)119863119863 119863 1199071199074119895119895+2 1199071199074119895119895+3 119907119907119899119899minus1 where 0 le 119895119895 119895 1198951198991198991198954119895 minus 1 for 119899119899 1198991 119899119899119899119899 4) 119863119863 119863 1199071199074119895119895+2 1199071199074119895119895+3 119907119907119899119899minus2 119907119907119899119899minus1 where 0 le 119895119895 119895 1198951198991198991198954119895for 119899119899 119899 2 3 1198991198991198991198994) are 119899119899-sets of 119899119899prime119899119875119875119899119899) en 119898119898119899119899119899prime119899119875119875119899119899) minus119863119863) 119863 6

erefore

11986311986311986311986310076511007651119899119899prime 100764910076491198751198751198991198991007665100766510076651007665 le 119899119899 10076511007651119899119899prime 100764910076491198751198751198991198991007665100766510076651007665 + 6 (22)

Now we claim that if 119898119898119899119898119898 minus 119899119899) 119898 6 for any dominating set 119899119899other than119863119863 then

119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911987511987511989911989910076651007665 minus 11989911989910076651007665 ge 119899119899 10076511007651119899119899prime 100764910076491198751198751198991198991007665100766510076651007665 + 6 (23)

Case 1 If 1198991198991 is any dominating set other than 119863119863 and119898119898119899119899119899prime119899119875119875119899119899)minus1198991198991) gt 4 or 5 then 1198781198991198991119878 ge 119899119899119899119899119899

prime119899119875119875119899119899))+2 Consequently1198781198991198991119878 + 119898119898119899119899119899

prime119899119875119875119899119899) minus 1198991198991) ge 119899119899119899119899119899prime119899119875119875119899119899)) + 6

Case 2 If 1198991198992 is any dominating set other than 119863119863 and 1198991198991 of119899119899prime119899119875119875119899119899) with 119898119898119899119899119899prime119899119875119875119899119899) minus 1198991198992) 119863 3 then 1198781198991198992119878 ge 119899119899119899 minus 1198991198991198991198953119899)Consequently 1198781198991198992119878 + 119898119898119899119899119899prime119899119875119875119899119899) minus 1198991198992) ge 119899119899119899 minus 1198991198991198991198953119899) + 3 gt1198991198991198991198991198952119899 + 1) + 6 119863 119899119899119899119899119899prime119899119875119875119899119899)) + 6

Case 3 If 1198991198993 is any dominating set other than119863119863 1198991198991 and 1198991198992 of119899119899prime119899119875119875119899119899)with119898119898119899119899119899

prime119899119875119875119899119899)minus1198991198993) 119863 2 then 1198781198991198993119878 ge 119899119899minus2 Consequently1198781198991198993119878 + 119898119898119899119899119899

prime119899119875119875119899119899) minus 1198991198993) ge 119899119899119899 minus 2) + 2 119863 119899119899 gt 1198991198991198991198991198952119899 + 1) + 6 119863119899119899119899119899119899prime119899119875119875119899119899)) + 6

Case 4 If 1198991198994 is any dominating set other than119863119863 1198991198991 1198991198992 and 1198991198993of 119899119899prime119899119875119875119899119899)with119898119898119899119899119899

prime119899119875119875119899119899)minus1198991198994) 119863 1 then 1198781198991198994119878 ge 119899119899 Consequently1198781198991198994119878 +119898119898119899119899119899

prime119899119875119875119899119899)minus1198991198994) ge 119899119899+1 gt 1198991198991198991198991198952119899+1)+6 119863 119899119899119899119899119899prime119899119875119875119899119899))+6

us from (22) and (23)we have119863119863119863119863119899119899119899prime119899119875119875119899119899)) 119863 119899119899119899119899119899prime119899119875119875119899119899))+

6

eorem 10 For all 119899119899 ge 3

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076651007665 119863

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899119899119899119899 4)

119899119899 + 12

if 119899119899 119899 1 3 119899119899119899119899 4)

119899119899 + 22

if 119899119899 119899 2 119899119899119899119899 4)

(24)

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119899119899prime119899119862119862119899119899) Now either of the vertexfrom 119907119907119894119894minus1 and 119907119907119894119894+1 must belong to any dominating set 119899119899 as119873119873119899119906119906119894119894) 119863 119907119907119894119894minus1 119907119907119894119894+1

Consequently

119878119899119899119878 ge1198991198992for any dominating set 119899119899 (25)

Now depending upon the different possibilities of 119899119899 wechoose 119899119899 as follows

For 119899119899 119899 0 119899119899119899119899 4) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 with 119878119899119899119878 1198631198991198991198952 (where 1 le 119895119895 119895 1198951198991198991198954119895)for 119899119899 119899 1 119899119899119899119899 4) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 with 119878119899119899119878 119863 119899119899119899 +1)1198952 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 2 1198991198991198991198994) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 with 119878119899119899119878 119863119899119899119899 + 2)1198952 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 3 1198991198991198991198994) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 with 119878119899119899119878 119863119899119899119899 + 1)1198952 (where 1 le 119895119895 le 1198951198991198991198954119895)

We claim that each 119899119899 is a minimal dominating setof 119899119899prime119899119862119862119899119899) since 1198731198731198991199071199071) 119863 1199071199072 119907119907119899119899 1199061199062 119906119906119899119899 1198731198731198991199071199074119895119895) 1198631199071199074119895119895minus1 1199071199074119895119895+1 1199061199064119895119895minus1 1199061199064119895119895+1 1198731198731198991199071199074119895119895+1) 119863 1199071199074119895119895 1199071199074119895119895+2 1199061199064119895119895 1199061199064119895119895+2and 119873119873119899119907119907119899119899) 119863 1199071199071 119907119907119899119899minus1 1199061199061 119906119906119899119899minus1 and removal of 1199071199074119895119895 a vertex1199061199064119895119895+1 will not be dominated by any vertex hence from (25)

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076651007665 119863

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899119899119899119899 4)

119899119899 + 12

if 119899119899 119899 1 3 119899119899119899119899 4)

119899119899 + 22

if 119899119899 119899 2 119899119899119899119899 4)

(26)

eorem 11

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076651007665 119863100761810076181007626100762610076421007642

119899119899 + 1119899 3 le 119899119899 le 6

119899119899 minus 10078121007812119899119899310078281007828 + 3119899 7 le 119899119899 le 18

(27)

ISRN Combinatorics 5

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119878119878prime(119862119862119899119899)

Case 1 (119899119899 119899 119899) From eorem 10 120574120574(119878119878prime(119862119862119899)) 119899 2 and 119863119863 1198991199071199071 1199071199072 is a 120574120574-set of 119878119878

prime(119862119862119899) then119898119898(119878119878prime(119862119862119899) minus119863119863) 119899 119899 erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198621198621198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 11986311986310076651007665

119899 2 + 119899 119899 5(28)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(119862119862119899) with119898119898(119878119878prime(119862119862119899) minus 1198781198781) 119899 2 then |1198781198781| ge 119899 is implies that

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 119878119878110076651007665 ge 2 + 119899 119899 5 (29)

Let 1198781198782 119899 1199071199071 1199071199072 119907119907119899 be dominating set of 119878119878prime(119862119862119899) and119898119898(119878119878prime(119862119862119899) minus 1198781198782) 119899 1 is implies that

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 119878119878210076651007665 119899 119899 + 1 119899 4 (30)

Hence from (28) (29) and (30)119863119863119863119863(119878119878prime119862119862119899) 119899 4

Case 2 (119899119899 119899 4) From eorem 10 120574120574(119878119878prime(1198621198624)) 119899 2 and 119863119863 1198991199071199071 1199071199072 is a 120574120574-set of 119878119878

prime(1198621198624) then119898119898(119878119878prime(1198621198624) minus 119863119863) 119899 119899

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986241007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986241007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862410076651007665 minus 11986311986310076651007665

119899 2 + 119899 119899 5(31)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198624) and119898119898(119878119878prime(1198621198624) minus 1198781198781) 119899 2 or 1 then |1198781198781| ge 4 erefore

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862410076651007665 minus 119878119878110076651007665 ge 4 + 2 119899 6 (32)

Hence from (31) and (32)119863119863119863119863(119878119878prime(1198621198624)) 119899 5

Case 3 (119899119899 119899 5) From eorem 10 120574120574(119878119878prime(1198621198625)) 119899 119899 and 119863119863 1198991199071199072 119907119907119899 1199071199074 is a 120574120574-set of 119878119878

prime(1198621198625) then119898119898(119878119878prime(1198621198625) minus 119863119863) 119899 6

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 11986311986310076651007665

119899 119899 + 6 119899 9(33)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198625) and119898119898(119878119878prime(1198621198625) minus 1198781198781) 119899 4 or 5 then |1198781198781| ge 120574120574(119878119878

prime(1198621198625)) 119899 119899is implies

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 119878119878110076651007665 ge 120574120574 10076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 + 4

119899 119899 + 4 119899 7(34)

If 1198781198782 is any dominating set other than 119863119863 and 1198781198781 with119898119898(119878119878prime(1198621198625) minus 1198781198782) 119899 2 or 119899 then |1198781198782| ge 5 is implies that

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 119878119878210076651007665 ge 5 + 2 119899 7 (35)

Let 119878119878119899 119899 1199071199071 1199071199072 119907119907119899 1199071199074 1199071199075 be dominating set of 119878119878prime(1198621198625) then119898119898(119878119878prime(1198621198625) minus 119878119878119899) 119899 1 is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 11987811987811989910076651007665 119899 5 + 1 119899 6 (36)

Hence from (33) (34) (35) and (36)119863119863119863119863(119878119878prime(1198621198625)) 119899 6

Case 4 (119899119899 119899 6) From eorem 10 120574120574(119878119878prime(1198621198626)) 119899 4 and 119863119863 1198991199071199071 1199071199072 1199071199074 1199071199075 is a 120574120574-set of 119878119878

prime(1198621198626) then119898119898(119878119878prime(1198621198626) minus 119863119863) 119899 119899

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986261007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986261007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 11986311986310076651007665

119899 4 + 119899 119899 7(37)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198626) and119898119898(119878119878prime(1198621198626) minus 1198781198781) 119899 2 then |1198781198781| ge 6 is implies that

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 119878119878110076651007665 ge 6 + 2 119899 8 (38)

Let 1198781198782 119899 1199071199071 1199071199072 119907119907119899 1199071199074 1199071199075 1199071199076 be dominating set of 119878119878prime(1198621198626)then119898119898(119878119878prime(1198621198626) minus 1198781198782) 119899 1

erefore

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 119878119878210076651007665 119899 6 + 1 119899 7 (39)

Hence from (37) (38) and (39)119863119863119863119863(119878119878prime(1198621198626)) 119899 7

Case 5 (7 le 119899119899 le 18) We know that

11986311986311986311986310076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11986311986310076651007665 (40)

e domination number with 120574120574(119878119878prime(119862119862119899119899)) +119898119898(119878119878prime(119862119862119899119899) minus119863119863) and

set 119878119878 with |119878119878| + 119898119898(119878119878prime(119862119862119899119899) minus 119878119878) are shown in Table 2From Table 2 for any dominating set 119878119878 other than 119863119863 of

119878119878prime(119862119862119899119899) we have

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11987811987810076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11986311986310076651007665

(41)

If 1198781198781 is any dominating set of 119878119878prime(119862119862119899119899) other than119863119863 and 119878119878 with119898119898(119878119878prime(119862119862119899119899) minus 1198781198781) 119899 4 or 5 then |119878119878| le |1198781198781|

is implies that

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11987811987810076651007665 le 10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 119878119878110076651007665 (42)

If 1198781198782 is any dominating set of 119878119878prime(119862119862119899119899) other than 119863119863 119878119878 and 1198781198781with119898119898(119878119878prime(119862119862119899119899) minus 1198781198782) 119899 2 or 1 then |1198781198782| ge 119899119899

6 ISRN Combinatorics

T 2

119899119899 119899119899119899119899119899prime119899119862119862119899119899)) 119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) 119899119899119899119899119899prime119899119862119862119899119899)) + 119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) 119899119899 119898119898119899119899119899prime119899119862119862119899119899) minus 119899119899) 119878119899119899119878 + 119898119898119899119899119899prime119899119862119862119899119899) minus 119899119899)

7 4 6 10 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 3 88 4 6 10 1199071199071 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 3 99 5 6 11 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 3 910 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 3 1011 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 3 1112 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990712 3 1113 7 6 13 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 3 1214 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1315 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1316 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 11990711990716 3 14

is implies that

10092491009249119899119899210092491009249 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 119899119899210076671007667 ge 119899119899 + 1119899 (43)

Hence from Table 2 and the results (40) to (43) we concludethat

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 = 119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 11989911989910076671007667

= 119899119899 minus 10078121007812119899119899310078281007828 + 3119899

(44)

Hence the result

eorem 12 For all 119899119899 ge 17 119863119863119863119863119899119899119899prime119899119862119862119899119899)) = 119899119899119899119899119899prime119862119862119899119899) + 6

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119899119899prime119899119862119862119899119899) en fromeorem 10

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 =

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899mod 4)

119899119899 + 12

if 119899119899 119899 1 3 119899mod 4)

119899119899 + 22

if 119899119899 119899 2 119899mod 4)

(45)

119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 (where 1 le 119895119895 119895 1198951198991198991198954119895) for 119899119899 1198990 119899mod4) 119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 (where 1 le 119895119895 le 1198951198991198991198954119895) for119899119899 119899 1 119899mod 4) 119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 2 119899mod4) or 119899119899 119899 3 119899mod4) are 119899119899-sets of 119899119899prime119899119862119862119899119899)en119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) = 6

erefore

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 le 119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 + 6119899 (46)

Now we claim that if 119898119898119899119899119899prime119899119862119862119899119899)) ne 6 for any dominating set 119899119899other than119863119863 then

119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 11989911989910076671007667 ge 119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 + 6119899 (47)

Case 1 If 1198991198991 is any dominating set other than 119863119863 and119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198991) = 4 or 5 then 1198781198991198991119878 ge 119899119899119899119899119899prime119899119862119862119899119899)) + 2Consequently 1198781198991198991119878 + 119898119898119899119899119899

prime119899119862119862119899119899) minus 1198991198991) ge 119899119899119899119899119899prime119899119862119862119899119899)) + 6

Case 2 If 1198991198992 is any dominating set other than 119863119863 and 1198991198991 of119899119899prime119899119862119862119899119899) with 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198992) = 3 then 1198781198991198992119878 ge 119899119899119899 minus 1198951198991198991198953119895)Consequently 1198781198991198992119878 + 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198992) ge 119899119899119899 minus 1198951198991198991198953119895) + 3 119899119899119899119899119899119899prime119899119862119862119899119899)) + 6

Case 3 If 1198991198993 is any dominating set other than 119863119863 1198991198991 and1198991198992 of 119899119899prime119899119862119862119899119899) with 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198993) = 2 or 1 then 1198781198991198993119878 ge 119899119899Consequently 1198781198991198993119878 + 119898119898119899119899119899

prime119899119862119862119899119899) minus 1198991198993) ge 119899119899 + 1 119899 119899119899119899119899119899prime119899119862119862119899119899)) + 6

us from (46) and (47) we have119863119863119863119863119899119899119899prime119899119862119862119899119899))

3 Concluding Remarks

e domination and vulnerability of network are two impor-tant concepts for the network security We have studied animportant measure of vulnerability known as dominationintegrity and investigate domination integrity of splittinggraphs of path and cycle e results reported here throwsome light in the direction to nd the domination integrityof larger graph obtained from the given graph

References

[1] C A Barefoot R Entringer and H C Swart ldquoVulnerabilityin Graphs-a Comparative Surveyrdquo Journal of CombinatorialMathematics and Combinatorial Computing vol 1 pp 13ndash221987

[2] C A Barefoot R Entringer and H C Swart ldquoIntegrity of treesand powers of cyclesrdquo Congressus Numerantium vol 58 pp103ndash114 1987

[3] W Goddard andH C Swart ldquoOn the integrity of combinationsof graphsrdquo Journal of CombinatorialMathematics andCombina-torial Computing vol 4 pp 3ndash18 1988

[4] W Goddard and H C Swart ldquoIntegrity in graphs bounds andbasicsrdquo Journal of Combinatorial Mathematics and Combinato-rial Computing vol 7 pp 139ndash151 1990

[5] A Mamut and E Vumar A Note on the Integrity of MiddleGraphs Lecture Notes in Computer Science vol 4381 SpringerNew York NY USA 2007

ISRN Combinatorics 7

[6] P Duumlndar and A Aytaccedil ldquoIntegrity of total graphs via certainparametersrdquoMathematical Notes vol 76 no 5-6 pp 665ndash6722004

[7] R Sundareswaran and V Swaminathan ldquoDomination Integrityin graphsrdquo in Proceedings of International conference on Mathe-matical and Experimental Physics pp 46ndash57 Narosa PublishingHouse 2010

[8] R Sundareswaran and V Swaminathan ldquoDomination integrityof middle graphsrdquo in Algebra Graph eory and eir Applica-tions T T Chelvam S Somasundaram and R Kala Eds pp88ndash92 Narosa Publishing House Delhi India 2010

[9] S Vaidya and N Kothari ldquoSome new results on dominationintegrity of graphsrdquo Open Journal of Discrete Mathematics vol2 pp 96ndash98 2012

[10] D B West Introduction To Graph eory 2e Prentice-HallNew Delhi India 2003

[11] T Haynes S Hedetniemi and P Slater Fundamentals ofDomination in Graphs Marcel Dekker New York NY USA1998

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

ISRN Combinatorics 3

erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11986311986310076651007665

= 119899119899 minus 119899 + 119899 = 119899119899119899(6)

If 119878119878 is any dominating set other than 119863119863 of 119878119878prime(119875119875119899119899) with119898119898(119878119878prime(119875119875119899119899) minus 119878119878) = 119878 then |119878119878| 119878 119899119899 is implies that

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987810076651007665 119878 119899119899 + 119878 119899 119899119899119899 (7)

Hence from (6) and (7)119863119863119863119863(119878119878prime(119875119875119899119899)) = 119899119899

Subcase (iii) (119899119899 = 119899) Fromeorem 7 120574120574(119878119878prime(119875119875119899)) = 4 and119863119863 =119907119907119899 1199071199073 1199071199075 1199071199076 is a 120574120574-set of 119878119878

prime(119875119875119899) en119898119898(119878119878prime(119875119875119899) minus 119863119863) = 3

erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989910076651007665 minus 11986311986310076651007665 = 4 + 3 = 119899119899

(8)

Now if 119878119878119878 is any dominating set other than 119863119863 of 119878119878prime(119875119875119899) and119898119898(119878119878prime(119875119875119899) minus 119878119878119878) = 119899 then |119878119878119878| 119878 5 is implies that

1009249100924911987811987811987810092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989910076651007665 minus 11987811987811987810076651007665 119878 5 + 119899 = 119899119899 (9)

Also if 119878119878119899 is any dominating set other than119863119863 and 119878119878119878 of 119878119878prime(119875119875119899)

with119898119898(119878119878prime(119875119875119899) minus 119878119878119899) = 119878 then |119878119878119899| 119878 119899is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989910076651007665 minus 11987811987811989910076651007665 119878 119899 + 119878 = 8119899 (10)

Hence from (8) (9) and (10)119863119863119863119863(119878119878prime(119875119875119899)) = 119899

Case 3 (119899119899 = 119878119899) From eorem 7 120574120574(119878119878prime(119875119875119878119899)) = 6 and 119863119863 =119907119907119899 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 is a 120574120574-set of 119878119878

prime(119875119875119878119899) en 119898119898(119878119878prime(119875119875119878119899) minus119863119863) = 3

erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198781198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198781198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511987811989910076651007665 minus 11986311986310076651007665

= 6 + 3 = 9119899(11)

Now if 119878119878119878 is any dominating set other than 119863119863 of 119878119878prime(119875119875119878119899) and119898119898(119878119878prime(119875119875119878119899) minus 119878119878119878) = 119899 then |119878119878119878| 119878 8

is implies that

1009249100924911987811987811987810092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511987811989910076651007665 minus 11987811987811987810076651007665 119878 8 + 119899 = 119878119899119899 (12)

Also if 119878119878119899 is any dominating set other than119863119863 and 119878119878119878 of 119878119878prime(119875119875119878119899)

with119898119898(119878119878prime(119875119875119878119899) minus 119878119878119899) = 119878 then |119878119878119899| 119878 119878119899is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511987811989910076651007665 minus 11987811987811989910076651007665 119878 119878119899 + 119878 = 119878119878119899 (13)

Hence from (11) (12) and (13)119863119863119863119863(119878119878prime(119875119875119878119899)) = 9

Case 4 (119899119899 = 8 9 and 119878119878 le 119899119899 le 1198789) We know that

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11986311986310076651007665 119899 (14)

e domination number with 120574120574(119878119878prime(119862119862119899119899)) +119898119898(119878119878prime(119875119875119899119899) minus119863119863) and

set 119878119878 with |119878119878| + 119898119898(119878119878prime(119875119875119899119899) minus 119878119878) are shown in Table 1From Table 1

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987810076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11986311986310076651007665 119899

(15)

If 119878119878119878 is any dominating set of 119878119878prime(119875119875119899119899) other than119863119863 and 119878119878 with119898119898(119878119878prime(119875119875119899119899) minus 119878119878119878) = 4 or 5 then |119878119878| le |119878119878119878|

is implies that

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987810076651007665 le 1009249100924911987811987811987810092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987811987810076651007665 119899 (16)

If 119878119878119899 is any dominating set other than 119863119863 119878119878119878 and 119878119878 with119898119898(119878119878prime(119875119875119899119899) minus 119878119878119899) = 119899 then |119878119878119899| 119878 119899119899 minus 119899

is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987811989910076651007665 119878 (119899119899 minus 119899) + 119899 = 119899119899119899 (17)

If 1198781198783 is any dominating set other than 119863119863 119878119878119878 119878119878119899 and 119878119878 with119898119898(119878119878prime(119875119875119899119899) minus 1198781198783) = 119878 then |1198781198783| 119878 119899119899

is implies that

10092491009249119878119878310092491009249 + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 119878119878310076651007665 119878 119899119899 + 119878119899 (18)

Hence from Table 1 and the results (14) to (18) we concludethat

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 = |119878119878| + 11989811989810076511007651119878119878prime 1007649100764911987511987511989911989910076651007665 minus 11987811987810076651007665

=

10076181007618100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076261007626100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

3 if 119899119899 = 119899119899119899 if 3 ⩽ 119899119899 ⩽ 9119899119899 minus 119878 if 119878119899 ⩽ 119899119899 ⩽ 119878119899119899119899 minus 119899 if 1198783 ⩽ 119899119899 ⩽ 1198785119899119899 minus 3 if 1198786 ⩽ 119899119899 ⩽ 11987881198785 if 119899119899 = 1198789119899

(19)

Hence the result

eorem 9

11986311986311986311986310076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 = 120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 + 6 forall119899119899 119878 119899119899119899 (20)

Proof Let 119907119907119878 119907119907119899hellip 119907119907119899119899 be the vertices of path 119875119875119899119899 and119906119906119878 119906119906119899hellip 119906119906119899119899 be the vertices corresponding to 119907119907119878 119907119907119899hellip 119907119907119899119899which are added to obtain 119878119878prime(119875119875119899119899) en fromeorem 7

120574120574 10076511007651119878119878prime 100764910076491198751198751198991198991007665100766510076651007665 =

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

119899119899119899

if 119899119899 119899 119899 (mod 4)

119899119899 + 119878119899

if 119899119899 119899 119878 3 (mod 4)

119899119899 + 119899119899

if 119899119899 119899 119899 (mod 4)

(21)

4 ISRN Combinatorics

T 1

119899119899 119899119899119899119899119899prime119899119875119875119899119899)) 119898119898119899119899119899prime119899119875119875119899119899) minus 119863119863) 119899119899119899119899119899prime119899119875119875119899119899)) + 119898119898119899119899119899prime119899119875119875119899119899) minus 119863119863) 119899119899 119898119898119899119899119899prime119899119875119875119899119899) minus 119899119899) 119878119899119899119878 + 119898119898119899119899119899prime119899119875119875119899119899) minus 119899119899)

8 4 6 10 1199071199072 1199071199073 1199071199074 1199071199076 1199071199077 3 89 5 6 11 1199071199072 1199071199073 1199071199075 1199071199076 1199071199077 1199071199078 3 911 6 6 12 1199071199072 1199071199073 1199071199075 1199071199076 1199071199077 1199071199079 11990711990710 3 1012 6 6 12 1199071199072 1199071199073 1199071199075 1199071199076 1199071199077 1199071199078 11990711990710 11990711990711 3 1113 7 6 13 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 3 1114 8 6 14 1199071199072 1199071199073 1199071199074 1199071199076 1199071199077 1199071199079 11990711990710 11990711990712 11990711990713 3 1215 8 6 14 1199071199072 1199071199073 1199071199074 1199071199076 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1316 8 6 14 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 3 1317 9 6 15 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 11990711990716 3 1418 10 6 16 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 11990711990716 11990711990717 3 1519 10 6 16 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 11990711990717 11990711990718 3 15

119863119863 119863 1199071199074119895119895+2 1199071199074119895119895+3 where 0 le 119895119895 119895 1198951198991198991198954119895 for 119899119899 119899 0 119899119899119899119899 4)119863119863 119863 1199071199074119895119895+2 1199071199074119895119895+3 119907119907119899119899minus1 where 0 le 119895119895 119895 1198951198991198991198954119895 minus 1 for 119899119899 1198991 119899119899119899119899 4) 119863119863 119863 1199071199074119895119895+2 1199071199074119895119895+3 119907119907119899119899minus2 119907119907119899119899minus1 where 0 le 119895119895 119895 1198951198991198991198954119895for 119899119899 119899 2 3 1198991198991198991198994) are 119899119899-sets of 119899119899prime119899119875119875119899119899) en 119898119898119899119899119899prime119899119875119875119899119899) minus119863119863) 119863 6

erefore

11986311986311986311986310076511007651119899119899prime 100764910076491198751198751198991198991007665100766510076651007665 le 119899119899 10076511007651119899119899prime 100764910076491198751198751198991198991007665100766510076651007665 + 6 (22)

Now we claim that if 119898119898119899119898119898 minus 119899119899) 119898 6 for any dominating set 119899119899other than119863119863 then

119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911987511987511989911989910076651007665 minus 11989911989910076651007665 ge 119899119899 10076511007651119899119899prime 100764910076491198751198751198991198991007665100766510076651007665 + 6 (23)

Case 1 If 1198991198991 is any dominating set other than 119863119863 and119898119898119899119899119899prime119899119875119875119899119899)minus1198991198991) gt 4 or 5 then 1198781198991198991119878 ge 119899119899119899119899119899

prime119899119875119875119899119899))+2 Consequently1198781198991198991119878 + 119898119898119899119899119899

prime119899119875119875119899119899) minus 1198991198991) ge 119899119899119899119899119899prime119899119875119875119899119899)) + 6

Case 2 If 1198991198992 is any dominating set other than 119863119863 and 1198991198991 of119899119899prime119899119875119875119899119899) with 119898119898119899119899119899prime119899119875119875119899119899) minus 1198991198992) 119863 3 then 1198781198991198992119878 ge 119899119899119899 minus 1198991198991198991198953119899)Consequently 1198781198991198992119878 + 119898119898119899119899119899prime119899119875119875119899119899) minus 1198991198992) ge 119899119899119899 minus 1198991198991198991198953119899) + 3 gt1198991198991198991198991198952119899 + 1) + 6 119863 119899119899119899119899119899prime119899119875119875119899119899)) + 6

Case 3 If 1198991198993 is any dominating set other than119863119863 1198991198991 and 1198991198992 of119899119899prime119899119875119875119899119899)with119898119898119899119899119899

prime119899119875119875119899119899)minus1198991198993) 119863 2 then 1198781198991198993119878 ge 119899119899minus2 Consequently1198781198991198993119878 + 119898119898119899119899119899

prime119899119875119875119899119899) minus 1198991198993) ge 119899119899119899 minus 2) + 2 119863 119899119899 gt 1198991198991198991198991198952119899 + 1) + 6 119863119899119899119899119899119899prime119899119875119875119899119899)) + 6

Case 4 If 1198991198994 is any dominating set other than119863119863 1198991198991 1198991198992 and 1198991198993of 119899119899prime119899119875119875119899119899)with119898119898119899119899119899

prime119899119875119875119899119899)minus1198991198994) 119863 1 then 1198781198991198994119878 ge 119899119899 Consequently1198781198991198994119878 +119898119898119899119899119899

prime119899119875119875119899119899)minus1198991198994) ge 119899119899+1 gt 1198991198991198991198991198952119899+1)+6 119863 119899119899119899119899119899prime119899119875119875119899119899))+6

us from (22) and (23)we have119863119863119863119863119899119899119899prime119899119875119875119899119899)) 119863 119899119899119899119899119899prime119899119875119875119899119899))+

6

eorem 10 For all 119899119899 ge 3

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076651007665 119863

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899119899119899119899 4)

119899119899 + 12

if 119899119899 119899 1 3 119899119899119899119899 4)

119899119899 + 22

if 119899119899 119899 2 119899119899119899119899 4)

(24)

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119899119899prime119899119862119862119899119899) Now either of the vertexfrom 119907119907119894119894minus1 and 119907119907119894119894+1 must belong to any dominating set 119899119899 as119873119873119899119906119906119894119894) 119863 119907119907119894119894minus1 119907119907119894119894+1

Consequently

119878119899119899119878 ge1198991198992for any dominating set 119899119899 (25)

Now depending upon the different possibilities of 119899119899 wechoose 119899119899 as follows

For 119899119899 119899 0 119899119899119899119899 4) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 with 119878119899119899119878 1198631198991198991198952 (where 1 le 119895119895 119895 1198951198991198991198954119895)for 119899119899 119899 1 119899119899119899119899 4) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 with 119878119899119899119878 119863 119899119899119899 +1)1198952 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 2 1198991198991198991198994) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 with 119878119899119899119878 119863119899119899119899 + 2)1198952 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 3 1198991198991198991198994) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 with 119878119899119899119878 119863119899119899119899 + 1)1198952 (where 1 le 119895119895 le 1198951198991198991198954119895)

We claim that each 119899119899 is a minimal dominating setof 119899119899prime119899119862119862119899119899) since 1198731198731198991199071199071) 119863 1199071199072 119907119907119899119899 1199061199062 119906119906119899119899 1198731198731198991199071199074119895119895) 1198631199071199074119895119895minus1 1199071199074119895119895+1 1199061199064119895119895minus1 1199061199064119895119895+1 1198731198731198991199071199074119895119895+1) 119863 1199071199074119895119895 1199071199074119895119895+2 1199061199064119895119895 1199061199064119895119895+2and 119873119873119899119907119907119899119899) 119863 1199071199071 119907119907119899119899minus1 1199061199061 119906119906119899119899minus1 and removal of 1199071199074119895119895 a vertex1199061199064119895119895+1 will not be dominated by any vertex hence from (25)

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076651007665 119863

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899119899119899119899 4)

119899119899 + 12

if 119899119899 119899 1 3 119899119899119899119899 4)

119899119899 + 22

if 119899119899 119899 2 119899119899119899119899 4)

(26)

eorem 11

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076651007665 119863100761810076181007626100762610076421007642

119899119899 + 1119899 3 le 119899119899 le 6

119899119899 minus 10078121007812119899119899310078281007828 + 3119899 7 le 119899119899 le 18

(27)

ISRN Combinatorics 5

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119878119878prime(119862119862119899119899)

Case 1 (119899119899 119899 119899) From eorem 10 120574120574(119878119878prime(119862119862119899)) 119899 2 and 119863119863 1198991199071199071 1199071199072 is a 120574120574-set of 119878119878

prime(119862119862119899) then119898119898(119878119878prime(119862119862119899) minus119863119863) 119899 119899 erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198621198621198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 11986311986310076651007665

119899 2 + 119899 119899 5(28)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(119862119862119899) with119898119898(119878119878prime(119862119862119899) minus 1198781198781) 119899 2 then |1198781198781| ge 119899 is implies that

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 119878119878110076651007665 ge 2 + 119899 119899 5 (29)

Let 1198781198782 119899 1199071199071 1199071199072 119907119907119899 be dominating set of 119878119878prime(119862119862119899) and119898119898(119878119878prime(119862119862119899) minus 1198781198782) 119899 1 is implies that

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 119878119878210076651007665 119899 119899 + 1 119899 4 (30)

Hence from (28) (29) and (30)119863119863119863119863(119878119878prime119862119862119899) 119899 4

Case 2 (119899119899 119899 4) From eorem 10 120574120574(119878119878prime(1198621198624)) 119899 2 and 119863119863 1198991199071199071 1199071199072 is a 120574120574-set of 119878119878

prime(1198621198624) then119898119898(119878119878prime(1198621198624) minus 119863119863) 119899 119899

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986241007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986241007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862410076651007665 minus 11986311986310076651007665

119899 2 + 119899 119899 5(31)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198624) and119898119898(119878119878prime(1198621198624) minus 1198781198781) 119899 2 or 1 then |1198781198781| ge 4 erefore

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862410076651007665 minus 119878119878110076651007665 ge 4 + 2 119899 6 (32)

Hence from (31) and (32)119863119863119863119863(119878119878prime(1198621198624)) 119899 5

Case 3 (119899119899 119899 5) From eorem 10 120574120574(119878119878prime(1198621198625)) 119899 119899 and 119863119863 1198991199071199072 119907119907119899 1199071199074 is a 120574120574-set of 119878119878

prime(1198621198625) then119898119898(119878119878prime(1198621198625) minus 119863119863) 119899 6

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 11986311986310076651007665

119899 119899 + 6 119899 9(33)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198625) and119898119898(119878119878prime(1198621198625) minus 1198781198781) 119899 4 or 5 then |1198781198781| ge 120574120574(119878119878

prime(1198621198625)) 119899 119899is implies

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 119878119878110076651007665 ge 120574120574 10076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 + 4

119899 119899 + 4 119899 7(34)

If 1198781198782 is any dominating set other than 119863119863 and 1198781198781 with119898119898(119878119878prime(1198621198625) minus 1198781198782) 119899 2 or 119899 then |1198781198782| ge 5 is implies that

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 119878119878210076651007665 ge 5 + 2 119899 7 (35)

Let 119878119878119899 119899 1199071199071 1199071199072 119907119907119899 1199071199074 1199071199075 be dominating set of 119878119878prime(1198621198625) then119898119898(119878119878prime(1198621198625) minus 119878119878119899) 119899 1 is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 11987811987811989910076651007665 119899 5 + 1 119899 6 (36)

Hence from (33) (34) (35) and (36)119863119863119863119863(119878119878prime(1198621198625)) 119899 6

Case 4 (119899119899 119899 6) From eorem 10 120574120574(119878119878prime(1198621198626)) 119899 4 and 119863119863 1198991199071199071 1199071199072 1199071199074 1199071199075 is a 120574120574-set of 119878119878

prime(1198621198626) then119898119898(119878119878prime(1198621198626) minus 119863119863) 119899 119899

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986261007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986261007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 11986311986310076651007665

119899 4 + 119899 119899 7(37)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198626) and119898119898(119878119878prime(1198621198626) minus 1198781198781) 119899 2 then |1198781198781| ge 6 is implies that

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 119878119878110076651007665 ge 6 + 2 119899 8 (38)

Let 1198781198782 119899 1199071199071 1199071199072 119907119907119899 1199071199074 1199071199075 1199071199076 be dominating set of 119878119878prime(1198621198626)then119898119898(119878119878prime(1198621198626) minus 1198781198782) 119899 1

erefore

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 119878119878210076651007665 119899 6 + 1 119899 7 (39)

Hence from (37) (38) and (39)119863119863119863119863(119878119878prime(1198621198626)) 119899 7

Case 5 (7 le 119899119899 le 18) We know that

11986311986311986311986310076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11986311986310076651007665 (40)

e domination number with 120574120574(119878119878prime(119862119862119899119899)) +119898119898(119878119878prime(119862119862119899119899) minus119863119863) and

set 119878119878 with |119878119878| + 119898119898(119878119878prime(119862119862119899119899) minus 119878119878) are shown in Table 2From Table 2 for any dominating set 119878119878 other than 119863119863 of

119878119878prime(119862119862119899119899) we have

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11987811987810076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11986311986310076651007665

(41)

If 1198781198781 is any dominating set of 119878119878prime(119862119862119899119899) other than119863119863 and 119878119878 with119898119898(119878119878prime(119862119862119899119899) minus 1198781198781) 119899 4 or 5 then |119878119878| le |1198781198781|

is implies that

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11987811987810076651007665 le 10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 119878119878110076651007665 (42)

If 1198781198782 is any dominating set of 119878119878prime(119862119862119899119899) other than 119863119863 119878119878 and 1198781198781with119898119898(119878119878prime(119862119862119899119899) minus 1198781198782) 119899 2 or 1 then |1198781198782| ge 119899119899

6 ISRN Combinatorics

T 2

119899119899 119899119899119899119899119899prime119899119862119862119899119899)) 119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) 119899119899119899119899119899prime119899119862119862119899119899)) + 119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) 119899119899 119898119898119899119899119899prime119899119862119862119899119899) minus 119899119899) 119878119899119899119878 + 119898119898119899119899119899prime119899119862119862119899119899) minus 119899119899)

7 4 6 10 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 3 88 4 6 10 1199071199071 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 3 99 5 6 11 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 3 910 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 3 1011 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 3 1112 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990712 3 1113 7 6 13 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 3 1214 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1315 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1316 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 11990711990716 3 14

is implies that

10092491009249119899119899210092491009249 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 119899119899210076671007667 ge 119899119899 + 1119899 (43)

Hence from Table 2 and the results (40) to (43) we concludethat

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 = 119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 11989911989910076671007667

= 119899119899 minus 10078121007812119899119899310078281007828 + 3119899

(44)

Hence the result

eorem 12 For all 119899119899 ge 17 119863119863119863119863119899119899119899prime119899119862119862119899119899)) = 119899119899119899119899119899prime119862119862119899119899) + 6

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119899119899prime119899119862119862119899119899) en fromeorem 10

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 =

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899mod 4)

119899119899 + 12

if 119899119899 119899 1 3 119899mod 4)

119899119899 + 22

if 119899119899 119899 2 119899mod 4)

(45)

119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 (where 1 le 119895119895 119895 1198951198991198991198954119895) for 119899119899 1198990 119899mod4) 119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 (where 1 le 119895119895 le 1198951198991198991198954119895) for119899119899 119899 1 119899mod 4) 119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 2 119899mod4) or 119899119899 119899 3 119899mod4) are 119899119899-sets of 119899119899prime119899119862119862119899119899)en119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) = 6

erefore

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 le 119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 + 6119899 (46)

Now we claim that if 119898119898119899119899119899prime119899119862119862119899119899)) ne 6 for any dominating set 119899119899other than119863119863 then

119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 11989911989910076671007667 ge 119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 + 6119899 (47)

Case 1 If 1198991198991 is any dominating set other than 119863119863 and119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198991) = 4 or 5 then 1198781198991198991119878 ge 119899119899119899119899119899prime119899119862119862119899119899)) + 2Consequently 1198781198991198991119878 + 119898119898119899119899119899

prime119899119862119862119899119899) minus 1198991198991) ge 119899119899119899119899119899prime119899119862119862119899119899)) + 6

Case 2 If 1198991198992 is any dominating set other than 119863119863 and 1198991198991 of119899119899prime119899119862119862119899119899) with 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198992) = 3 then 1198781198991198992119878 ge 119899119899119899 minus 1198951198991198991198953119895)Consequently 1198781198991198992119878 + 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198992) ge 119899119899119899 minus 1198951198991198991198953119895) + 3 119899119899119899119899119899119899prime119899119862119862119899119899)) + 6

Case 3 If 1198991198993 is any dominating set other than 119863119863 1198991198991 and1198991198992 of 119899119899prime119899119862119862119899119899) with 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198993) = 2 or 1 then 1198781198991198993119878 ge 119899119899Consequently 1198781198991198993119878 + 119898119898119899119899119899

prime119899119862119862119899119899) minus 1198991198993) ge 119899119899 + 1 119899 119899119899119899119899119899prime119899119862119862119899119899)) + 6

us from (46) and (47) we have119863119863119863119863119899119899119899prime119899119862119862119899119899))

3 Concluding Remarks

e domination and vulnerability of network are two impor-tant concepts for the network security We have studied animportant measure of vulnerability known as dominationintegrity and investigate domination integrity of splittinggraphs of path and cycle e results reported here throwsome light in the direction to nd the domination integrityof larger graph obtained from the given graph

References

[1] C A Barefoot R Entringer and H C Swart ldquoVulnerabilityin Graphs-a Comparative Surveyrdquo Journal of CombinatorialMathematics and Combinatorial Computing vol 1 pp 13ndash221987

[2] C A Barefoot R Entringer and H C Swart ldquoIntegrity of treesand powers of cyclesrdquo Congressus Numerantium vol 58 pp103ndash114 1987

[3] W Goddard andH C Swart ldquoOn the integrity of combinationsof graphsrdquo Journal of CombinatorialMathematics andCombina-torial Computing vol 4 pp 3ndash18 1988

[4] W Goddard and H C Swart ldquoIntegrity in graphs bounds andbasicsrdquo Journal of Combinatorial Mathematics and Combinato-rial Computing vol 7 pp 139ndash151 1990

[5] A Mamut and E Vumar A Note on the Integrity of MiddleGraphs Lecture Notes in Computer Science vol 4381 SpringerNew York NY USA 2007

ISRN Combinatorics 7

[6] P Duumlndar and A Aytaccedil ldquoIntegrity of total graphs via certainparametersrdquoMathematical Notes vol 76 no 5-6 pp 665ndash6722004

[7] R Sundareswaran and V Swaminathan ldquoDomination Integrityin graphsrdquo in Proceedings of International conference on Mathe-matical and Experimental Physics pp 46ndash57 Narosa PublishingHouse 2010

[8] R Sundareswaran and V Swaminathan ldquoDomination integrityof middle graphsrdquo in Algebra Graph eory and eir Applica-tions T T Chelvam S Somasundaram and R Kala Eds pp88ndash92 Narosa Publishing House Delhi India 2010

[9] S Vaidya and N Kothari ldquoSome new results on dominationintegrity of graphsrdquo Open Journal of Discrete Mathematics vol2 pp 96ndash98 2012

[10] D B West Introduction To Graph eory 2e Prentice-HallNew Delhi India 2003

[11] T Haynes S Hedetniemi and P Slater Fundamentals ofDomination in Graphs Marcel Dekker New York NY USA1998

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4 ISRN Combinatorics

T 1

119899119899 119899119899119899119899119899prime119899119875119875119899119899)) 119898119898119899119899119899prime119899119875119875119899119899) minus 119863119863) 119899119899119899119899119899prime119899119875119875119899119899)) + 119898119898119899119899119899prime119899119875119875119899119899) minus 119863119863) 119899119899 119898119898119899119899119899prime119899119875119875119899119899) minus 119899119899) 119878119899119899119878 + 119898119898119899119899119899prime119899119875119875119899119899) minus 119899119899)

8 4 6 10 1199071199072 1199071199073 1199071199074 1199071199076 1199071199077 3 89 5 6 11 1199071199072 1199071199073 1199071199075 1199071199076 1199071199077 1199071199078 3 911 6 6 12 1199071199072 1199071199073 1199071199075 1199071199076 1199071199077 1199071199079 11990711990710 3 1012 6 6 12 1199071199072 1199071199073 1199071199075 1199071199076 1199071199077 1199071199078 11990711990710 11990711990711 3 1113 7 6 13 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 3 1114 8 6 14 1199071199072 1199071199073 1199071199074 1199071199076 1199071199077 1199071199079 11990711990710 11990711990712 11990711990713 3 1215 8 6 14 1199071199072 1199071199073 1199071199074 1199071199076 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1316 8 6 14 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 3 1317 9 6 15 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 11990711990716 3 1418 10 6 16 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 11990711990716 11990711990717 3 1519 10 6 16 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 1199071199079 11990711990711 11990711990712 11990711990714 11990711990715 11990711990717 11990711990718 3 15

119863119863 119863 1199071199074119895119895+2 1199071199074119895119895+3 where 0 le 119895119895 119895 1198951198991198991198954119895 for 119899119899 119899 0 119899119899119899119899 4)119863119863 119863 1199071199074119895119895+2 1199071199074119895119895+3 119907119907119899119899minus1 where 0 le 119895119895 119895 1198951198991198991198954119895 minus 1 for 119899119899 1198991 119899119899119899119899 4) 119863119863 119863 1199071199074119895119895+2 1199071199074119895119895+3 119907119907119899119899minus2 119907119907119899119899minus1 where 0 le 119895119895 119895 1198951198991198991198954119895for 119899119899 119899 2 3 1198991198991198991198994) are 119899119899-sets of 119899119899prime119899119875119875119899119899) en 119898119898119899119899119899prime119899119875119875119899119899) minus119863119863) 119863 6

erefore

11986311986311986311986310076511007651119899119899prime 100764910076491198751198751198991198991007665100766510076651007665 le 119899119899 10076511007651119899119899prime 100764910076491198751198751198991198991007665100766510076651007665 + 6 (22)

Now we claim that if 119898119898119899119898119898 minus 119899119899) 119898 6 for any dominating set 119899119899other than119863119863 then

119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911987511987511989911989910076651007665 minus 11989911989910076651007665 ge 119899119899 10076511007651119899119899prime 100764910076491198751198751198991198991007665100766510076651007665 + 6 (23)

Case 1 If 1198991198991 is any dominating set other than 119863119863 and119898119898119899119899119899prime119899119875119875119899119899)minus1198991198991) gt 4 or 5 then 1198781198991198991119878 ge 119899119899119899119899119899

prime119899119875119875119899119899))+2 Consequently1198781198991198991119878 + 119898119898119899119899119899

prime119899119875119875119899119899) minus 1198991198991) ge 119899119899119899119899119899prime119899119875119875119899119899)) + 6

Case 2 If 1198991198992 is any dominating set other than 119863119863 and 1198991198991 of119899119899prime119899119875119875119899119899) with 119898119898119899119899119899prime119899119875119875119899119899) minus 1198991198992) 119863 3 then 1198781198991198992119878 ge 119899119899119899 minus 1198991198991198991198953119899)Consequently 1198781198991198992119878 + 119898119898119899119899119899prime119899119875119875119899119899) minus 1198991198992) ge 119899119899119899 minus 1198991198991198991198953119899) + 3 gt1198991198991198991198991198952119899 + 1) + 6 119863 119899119899119899119899119899prime119899119875119875119899119899)) + 6

Case 3 If 1198991198993 is any dominating set other than119863119863 1198991198991 and 1198991198992 of119899119899prime119899119875119875119899119899)with119898119898119899119899119899

prime119899119875119875119899119899)minus1198991198993) 119863 2 then 1198781198991198993119878 ge 119899119899minus2 Consequently1198781198991198993119878 + 119898119898119899119899119899

prime119899119875119875119899119899) minus 1198991198993) ge 119899119899119899 minus 2) + 2 119863 119899119899 gt 1198991198991198991198991198952119899 + 1) + 6 119863119899119899119899119899119899prime119899119875119875119899119899)) + 6

Case 4 If 1198991198994 is any dominating set other than119863119863 1198991198991 1198991198992 and 1198991198993of 119899119899prime119899119875119875119899119899)with119898119898119899119899119899

prime119899119875119875119899119899)minus1198991198994) 119863 1 then 1198781198991198994119878 ge 119899119899 Consequently1198781198991198994119878 +119898119898119899119899119899

prime119899119875119875119899119899)minus1198991198994) ge 119899119899+1 gt 1198991198991198991198991198952119899+1)+6 119863 119899119899119899119899119899prime119899119875119875119899119899))+6

us from (22) and (23)we have119863119863119863119863119899119899119899prime119899119875119875119899119899)) 119863 119899119899119899119899119899prime119899119875119875119899119899))+

6

eorem 10 For all 119899119899 ge 3

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076651007665 119863

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899119899119899119899 4)

119899119899 + 12

if 119899119899 119899 1 3 119899119899119899119899 4)

119899119899 + 22

if 119899119899 119899 2 119899119899119899119899 4)

(24)

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119899119899prime119899119862119862119899119899) Now either of the vertexfrom 119907119907119894119894minus1 and 119907119907119894119894+1 must belong to any dominating set 119899119899 as119873119873119899119906119906119894119894) 119863 119907119907119894119894minus1 119907119907119894119894+1

Consequently

119878119899119899119878 ge1198991198992for any dominating set 119899119899 (25)

Now depending upon the different possibilities of 119899119899 wechoose 119899119899 as follows

For 119899119899 119899 0 119899119899119899119899 4) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 with 119878119899119899119878 1198631198991198991198952 (where 1 le 119895119895 119895 1198951198991198991198954119895)for 119899119899 119899 1 119899119899119899119899 4) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 with 119878119899119899119878 119863 119899119899119899 +1)1198952 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 2 1198991198991198991198994) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 with 119878119899119899119878 119863119899119899119899 + 2)1198952 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 3 1198991198991198991198994) 119899119899 119863 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 with 119878119899119899119878 119863119899119899119899 + 1)1198952 (where 1 le 119895119895 le 1198951198991198991198954119895)

We claim that each 119899119899 is a minimal dominating setof 119899119899prime119899119862119862119899119899) since 1198731198731198991199071199071) 119863 1199071199072 119907119907119899119899 1199061199062 119906119906119899119899 1198731198731198991199071199074119895119895) 1198631199071199074119895119895minus1 1199071199074119895119895+1 1199061199064119895119895minus1 1199061199064119895119895+1 1198731198731198991199071199074119895119895+1) 119863 1199071199074119895119895 1199071199074119895119895+2 1199061199064119895119895 1199061199064119895119895+2and 119873119873119899119907119907119899119899) 119863 1199071199071 119907119907119899119899minus1 1199061199061 119906119906119899119899minus1 and removal of 1199071199074119895119895 a vertex1199061199064119895119895+1 will not be dominated by any vertex hence from (25)

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076651007665 119863

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899119899119899119899 4)

119899119899 + 12

if 119899119899 119899 1 3 119899119899119899119899 4)

119899119899 + 22

if 119899119899 119899 2 119899119899119899119899 4)

(26)

eorem 11

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076651007665 119863100761810076181007626100762610076421007642

119899119899 + 1119899 3 le 119899119899 le 6

119899119899 minus 10078121007812119899119899310078281007828 + 3119899 7 le 119899119899 le 18

(27)

ISRN Combinatorics 5

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119878119878prime(119862119862119899119899)

Case 1 (119899119899 119899 119899) From eorem 10 120574120574(119878119878prime(119862119862119899)) 119899 2 and 119863119863 1198991199071199071 1199071199072 is a 120574120574-set of 119878119878

prime(119862119862119899) then119898119898(119878119878prime(119862119862119899) minus119863119863) 119899 119899 erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198621198621198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 11986311986310076651007665

119899 2 + 119899 119899 5(28)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(119862119862119899) with119898119898(119878119878prime(119862119862119899) minus 1198781198781) 119899 2 then |1198781198781| ge 119899 is implies that

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 119878119878110076651007665 ge 2 + 119899 119899 5 (29)

Let 1198781198782 119899 1199071199071 1199071199072 119907119907119899 be dominating set of 119878119878prime(119862119862119899) and119898119898(119878119878prime(119862119862119899) minus 1198781198782) 119899 1 is implies that

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 119878119878210076651007665 119899 119899 + 1 119899 4 (30)

Hence from (28) (29) and (30)119863119863119863119863(119878119878prime119862119862119899) 119899 4

Case 2 (119899119899 119899 4) From eorem 10 120574120574(119878119878prime(1198621198624)) 119899 2 and 119863119863 1198991199071199071 1199071199072 is a 120574120574-set of 119878119878

prime(1198621198624) then119898119898(119878119878prime(1198621198624) minus 119863119863) 119899 119899

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986241007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986241007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862410076651007665 minus 11986311986310076651007665

119899 2 + 119899 119899 5(31)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198624) and119898119898(119878119878prime(1198621198624) minus 1198781198781) 119899 2 or 1 then |1198781198781| ge 4 erefore

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862410076651007665 minus 119878119878110076651007665 ge 4 + 2 119899 6 (32)

Hence from (31) and (32)119863119863119863119863(119878119878prime(1198621198624)) 119899 5

Case 3 (119899119899 119899 5) From eorem 10 120574120574(119878119878prime(1198621198625)) 119899 119899 and 119863119863 1198991199071199072 119907119907119899 1199071199074 is a 120574120574-set of 119878119878

prime(1198621198625) then119898119898(119878119878prime(1198621198625) minus 119863119863) 119899 6

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 11986311986310076651007665

119899 119899 + 6 119899 9(33)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198625) and119898119898(119878119878prime(1198621198625) minus 1198781198781) 119899 4 or 5 then |1198781198781| ge 120574120574(119878119878

prime(1198621198625)) 119899 119899is implies

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 119878119878110076651007665 ge 120574120574 10076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 + 4

119899 119899 + 4 119899 7(34)

If 1198781198782 is any dominating set other than 119863119863 and 1198781198781 with119898119898(119878119878prime(1198621198625) minus 1198781198782) 119899 2 or 119899 then |1198781198782| ge 5 is implies that

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 119878119878210076651007665 ge 5 + 2 119899 7 (35)

Let 119878119878119899 119899 1199071199071 1199071199072 119907119907119899 1199071199074 1199071199075 be dominating set of 119878119878prime(1198621198625) then119898119898(119878119878prime(1198621198625) minus 119878119878119899) 119899 1 is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 11987811987811989910076651007665 119899 5 + 1 119899 6 (36)

Hence from (33) (34) (35) and (36)119863119863119863119863(119878119878prime(1198621198625)) 119899 6

Case 4 (119899119899 119899 6) From eorem 10 120574120574(119878119878prime(1198621198626)) 119899 4 and 119863119863 1198991199071199071 1199071199072 1199071199074 1199071199075 is a 120574120574-set of 119878119878

prime(1198621198626) then119898119898(119878119878prime(1198621198626) minus 119863119863) 119899 119899

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986261007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986261007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 11986311986310076651007665

119899 4 + 119899 119899 7(37)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198626) and119898119898(119878119878prime(1198621198626) minus 1198781198781) 119899 2 then |1198781198781| ge 6 is implies that

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 119878119878110076651007665 ge 6 + 2 119899 8 (38)

Let 1198781198782 119899 1199071199071 1199071199072 119907119907119899 1199071199074 1199071199075 1199071199076 be dominating set of 119878119878prime(1198621198626)then119898119898(119878119878prime(1198621198626) minus 1198781198782) 119899 1

erefore

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 119878119878210076651007665 119899 6 + 1 119899 7 (39)

Hence from (37) (38) and (39)119863119863119863119863(119878119878prime(1198621198626)) 119899 7

Case 5 (7 le 119899119899 le 18) We know that

11986311986311986311986310076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11986311986310076651007665 (40)

e domination number with 120574120574(119878119878prime(119862119862119899119899)) +119898119898(119878119878prime(119862119862119899119899) minus119863119863) and

set 119878119878 with |119878119878| + 119898119898(119878119878prime(119862119862119899119899) minus 119878119878) are shown in Table 2From Table 2 for any dominating set 119878119878 other than 119863119863 of

119878119878prime(119862119862119899119899) we have

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11987811987810076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11986311986310076651007665

(41)

If 1198781198781 is any dominating set of 119878119878prime(119862119862119899119899) other than119863119863 and 119878119878 with119898119898(119878119878prime(119862119862119899119899) minus 1198781198781) 119899 4 or 5 then |119878119878| le |1198781198781|

is implies that

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11987811987810076651007665 le 10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 119878119878110076651007665 (42)

If 1198781198782 is any dominating set of 119878119878prime(119862119862119899119899) other than 119863119863 119878119878 and 1198781198781with119898119898(119878119878prime(119862119862119899119899) minus 1198781198782) 119899 2 or 1 then |1198781198782| ge 119899119899

6 ISRN Combinatorics

T 2

119899119899 119899119899119899119899119899prime119899119862119862119899119899)) 119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) 119899119899119899119899119899prime119899119862119862119899119899)) + 119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) 119899119899 119898119898119899119899119899prime119899119862119862119899119899) minus 119899119899) 119878119899119899119878 + 119898119898119899119899119899prime119899119862119862119899119899) minus 119899119899)

7 4 6 10 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 3 88 4 6 10 1199071199071 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 3 99 5 6 11 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 3 910 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 3 1011 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 3 1112 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990712 3 1113 7 6 13 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 3 1214 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1315 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1316 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 11990711990716 3 14

is implies that

10092491009249119899119899210092491009249 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 119899119899210076671007667 ge 119899119899 + 1119899 (43)

Hence from Table 2 and the results (40) to (43) we concludethat

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 = 119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 11989911989910076671007667

= 119899119899 minus 10078121007812119899119899310078281007828 + 3119899

(44)

Hence the result

eorem 12 For all 119899119899 ge 17 119863119863119863119863119899119899119899prime119899119862119862119899119899)) = 119899119899119899119899119899prime119862119862119899119899) + 6

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119899119899prime119899119862119862119899119899) en fromeorem 10

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 =

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899mod 4)

119899119899 + 12

if 119899119899 119899 1 3 119899mod 4)

119899119899 + 22

if 119899119899 119899 2 119899mod 4)

(45)

119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 (where 1 le 119895119895 119895 1198951198991198991198954119895) for 119899119899 1198990 119899mod4) 119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 (where 1 le 119895119895 le 1198951198991198991198954119895) for119899119899 119899 1 119899mod 4) 119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 2 119899mod4) or 119899119899 119899 3 119899mod4) are 119899119899-sets of 119899119899prime119899119862119862119899119899)en119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) = 6

erefore

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 le 119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 + 6119899 (46)

Now we claim that if 119898119898119899119899119899prime119899119862119862119899119899)) ne 6 for any dominating set 119899119899other than119863119863 then

119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 11989911989910076671007667 ge 119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 + 6119899 (47)

Case 1 If 1198991198991 is any dominating set other than 119863119863 and119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198991) = 4 or 5 then 1198781198991198991119878 ge 119899119899119899119899119899prime119899119862119862119899119899)) + 2Consequently 1198781198991198991119878 + 119898119898119899119899119899

prime119899119862119862119899119899) minus 1198991198991) ge 119899119899119899119899119899prime119899119862119862119899119899)) + 6

Case 2 If 1198991198992 is any dominating set other than 119863119863 and 1198991198991 of119899119899prime119899119862119862119899119899) with 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198992) = 3 then 1198781198991198992119878 ge 119899119899119899 minus 1198951198991198991198953119895)Consequently 1198781198991198992119878 + 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198992) ge 119899119899119899 minus 1198951198991198991198953119895) + 3 119899119899119899119899119899119899prime119899119862119862119899119899)) + 6

Case 3 If 1198991198993 is any dominating set other than 119863119863 1198991198991 and1198991198992 of 119899119899prime119899119862119862119899119899) with 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198993) = 2 or 1 then 1198781198991198993119878 ge 119899119899Consequently 1198781198991198993119878 + 119898119898119899119899119899

prime119899119862119862119899119899) minus 1198991198993) ge 119899119899 + 1 119899 119899119899119899119899119899prime119899119862119862119899119899)) + 6

us from (46) and (47) we have119863119863119863119863119899119899119899prime119899119862119862119899119899))

3 Concluding Remarks

e domination and vulnerability of network are two impor-tant concepts for the network security We have studied animportant measure of vulnerability known as dominationintegrity and investigate domination integrity of splittinggraphs of path and cycle e results reported here throwsome light in the direction to nd the domination integrityof larger graph obtained from the given graph

References

[1] C A Barefoot R Entringer and H C Swart ldquoVulnerabilityin Graphs-a Comparative Surveyrdquo Journal of CombinatorialMathematics and Combinatorial Computing vol 1 pp 13ndash221987

[2] C A Barefoot R Entringer and H C Swart ldquoIntegrity of treesand powers of cyclesrdquo Congressus Numerantium vol 58 pp103ndash114 1987

[3] W Goddard andH C Swart ldquoOn the integrity of combinationsof graphsrdquo Journal of CombinatorialMathematics andCombina-torial Computing vol 4 pp 3ndash18 1988

[4] W Goddard and H C Swart ldquoIntegrity in graphs bounds andbasicsrdquo Journal of Combinatorial Mathematics and Combinato-rial Computing vol 7 pp 139ndash151 1990

[5] A Mamut and E Vumar A Note on the Integrity of MiddleGraphs Lecture Notes in Computer Science vol 4381 SpringerNew York NY USA 2007

ISRN Combinatorics 7

[6] P Duumlndar and A Aytaccedil ldquoIntegrity of total graphs via certainparametersrdquoMathematical Notes vol 76 no 5-6 pp 665ndash6722004

[7] R Sundareswaran and V Swaminathan ldquoDomination Integrityin graphsrdquo in Proceedings of International conference on Mathe-matical and Experimental Physics pp 46ndash57 Narosa PublishingHouse 2010

[8] R Sundareswaran and V Swaminathan ldquoDomination integrityof middle graphsrdquo in Algebra Graph eory and eir Applica-tions T T Chelvam S Somasundaram and R Kala Eds pp88ndash92 Narosa Publishing House Delhi India 2010

[9] S Vaidya and N Kothari ldquoSome new results on dominationintegrity of graphsrdquo Open Journal of Discrete Mathematics vol2 pp 96ndash98 2012

[10] D B West Introduction To Graph eory 2e Prentice-HallNew Delhi India 2003

[11] T Haynes S Hedetniemi and P Slater Fundamentals ofDomination in Graphs Marcel Dekker New York NY USA1998

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Stochastic AnalysisInternational Journal of

ISRN Combinatorics 5

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119878119878prime(119862119862119899119899)

Case 1 (119899119899 119899 119899) From eorem 10 120574120574(119878119878prime(119862119862119899)) 119899 2 and 119863119863 1198991199071199071 1199071199072 is a 120574120574-set of 119878119878

prime(119862119862119899) then119898119898(119878119878prime(119862119862119899) minus119863119863) 119899 119899 erefore

11986311986311986311986310076511007651119878119878prime 100764910076491198621198621198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 11986311986310076651007665

119899 2 + 119899 119899 5(28)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(119862119862119899) with119898119898(119878119878prime(119862119862119899) minus 1198781198781) 119899 2 then |1198781198781| ge 119899 is implies that

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 119878119878110076651007665 ge 2 + 119899 119899 5 (29)

Let 1198781198782 119899 1199071199071 1199071199072 119907119907119899 be dominating set of 119878119878prime(119862119862119899) and119898119898(119878119878prime(119862119862119899) minus 1198781198782) 119899 1 is implies that

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989910076651007665 minus 119878119878210076651007665 119899 119899 + 1 119899 4 (30)

Hence from (28) (29) and (30)119863119863119863119863(119878119878prime119862119862119899) 119899 4

Case 2 (119899119899 119899 4) From eorem 10 120574120574(119878119878prime(1198621198624)) 119899 2 and 119863119863 1198991199071199071 1199071199072 is a 120574120574-set of 119878119878

prime(1198621198624) then119898119898(119878119878prime(1198621198624) minus 119863119863) 119899 119899

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986241007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986241007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862410076651007665 minus 11986311986310076651007665

119899 2 + 119899 119899 5(31)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198624) and119898119898(119878119878prime(1198621198624) minus 1198781198781) 119899 2 or 1 then |1198781198781| ge 4 erefore

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862410076651007665 minus 119878119878110076651007665 ge 4 + 2 119899 6 (32)

Hence from (31) and (32)119863119863119863119863(119878119878prime(1198621198624)) 119899 5

Case 3 (119899119899 119899 5) From eorem 10 120574120574(119878119878prime(1198621198625)) 119899 119899 and 119863119863 1198991199071199072 119907119907119899 1199071199074 is a 120574120574-set of 119878119878

prime(1198621198625) then119898119898(119878119878prime(1198621198625) minus 119863119863) 119899 6

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 11986311986310076651007665

119899 119899 + 6 119899 9(33)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198625) and119898119898(119878119878prime(1198621198625) minus 1198781198781) 119899 4 or 5 then |1198781198781| ge 120574120574(119878119878

prime(1198621198625)) 119899 119899is implies

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 119878119878110076651007665 ge 120574120574 10076511007651119878119878prime 1007649100764911986211986251007665100766510076651007665 + 4

119899 119899 + 4 119899 7(34)

If 1198781198782 is any dominating set other than 119863119863 and 1198781198781 with119898119898(119878119878prime(1198621198625) minus 1198781198782) 119899 2 or 119899 then |1198781198782| ge 5 is implies that

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 119878119878210076651007665 ge 5 + 2 119899 7 (35)

Let 119878119878119899 119899 1199071199071 1199071199072 119907119907119899 1199071199074 1199071199075 be dominating set of 119878119878prime(1198621198625) then119898119898(119878119878prime(1198621198625) minus 119878119878119899) 119899 1 is implies that

1009249100924911987811987811989910092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862510076651007665 minus 11987811987811989910076651007665 119899 5 + 1 119899 6 (36)

Hence from (33) (34) (35) and (36)119863119863119863119863(119878119878prime(1198621198625)) 119899 6

Case 4 (119899119899 119899 6) From eorem 10 120574120574(119878119878prime(1198621198626)) 119899 4 and 119863119863 1198991199071199071 1199071199072 1199071199074 1199071199075 is a 120574120574-set of 119878119878

prime(1198621198626) then119898119898(119878119878prime(1198621198626) minus 119863119863) 119899 119899

erefore

11986311986311986311986310076511007651119878119878prime 1007649100764911986211986261007665100766510076651007665 le 120574120574 10076511007651119878119878prime 1007649100764911986211986261007665100766510076651007665 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 11986311986310076651007665

119899 4 + 119899 119899 7(37)

If 1198781198781 is any dominating set other than 119863119863 of 119878119878prime(1198621198626) and119898119898(119878119878prime(1198621198626) minus 1198781198781) 119899 2 then |1198781198781| ge 6 is implies that

10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 119878119878110076651007665 ge 6 + 2 119899 8 (38)

Let 1198781198782 119899 1199071199071 1199071199072 119907119907119899 1199071199074 1199071199075 1199071199076 be dominating set of 119878119878prime(1198621198626)then119898119898(119878119878prime(1198621198626) minus 1198781198782) 119899 1

erefore

10092491009249119878119878210092491009249 + 11989811989810076511007651119878119878prime 10076491007649119862119862610076651007665 minus 119878119878210076651007665 119899 6 + 1 119899 7 (39)

Hence from (37) (38) and (39)119863119863119863119863(119878119878prime(1198621198626)) 119899 7

Case 5 (7 le 119899119899 le 18) We know that

11986311986311986311986310076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11986311986310076651007665 (40)

e domination number with 120574120574(119878119878prime(119862119862119899119899)) +119898119898(119878119878prime(119862119862119899119899) minus119863119863) and

set 119878119878 with |119878119878| + 119898119898(119878119878prime(119862119862119899119899) minus 119878119878) are shown in Table 2From Table 2 for any dominating set 119878119878 other than 119863119863 of

119878119878prime(119862119862119899119899) we have

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11987811987810076651007665 le 120574120574 10076511007651119878119878prime 100764910076491198621198621198991198991007665100766510076651007665 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11986311986310076651007665

(41)

If 1198781198781 is any dominating set of 119878119878prime(119862119862119899119899) other than119863119863 and 119878119878 with119898119898(119878119878prime(119862119862119899119899) minus 1198781198781) 119899 4 or 5 then |119878119878| le |1198781198781|

is implies that

|119878119878| + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 11987811987810076651007665 le 10092491009249119878119878110092491009249 + 11989811989810076511007651119878119878prime 1007649100764911986211986211989911989910076651007665 minus 119878119878110076651007665 (42)

If 1198781198782 is any dominating set of 119878119878prime(119862119862119899119899) other than 119863119863 119878119878 and 1198781198781with119898119898(119878119878prime(119862119862119899119899) minus 1198781198782) 119899 2 or 1 then |1198781198782| ge 119899119899

6 ISRN Combinatorics

T 2

119899119899 119899119899119899119899119899prime119899119862119862119899119899)) 119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) 119899119899119899119899119899prime119899119862119862119899119899)) + 119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) 119899119899 119898119898119899119899119899prime119899119862119862119899119899) minus 119899119899) 119878119899119899119878 + 119898119898119899119899119899prime119899119862119862119899119899) minus 119899119899)

7 4 6 10 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 3 88 4 6 10 1199071199071 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 3 99 5 6 11 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 3 910 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 3 1011 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 3 1112 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990712 3 1113 7 6 13 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 3 1214 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1315 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1316 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 11990711990716 3 14

is implies that

10092491009249119899119899210092491009249 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 119899119899210076671007667 ge 119899119899 + 1119899 (43)

Hence from Table 2 and the results (40) to (43) we concludethat

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 = 119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 11989911989910076671007667

= 119899119899 minus 10078121007812119899119899310078281007828 + 3119899

(44)

Hence the result

eorem 12 For all 119899119899 ge 17 119863119863119863119863119899119899119899prime119899119862119862119899119899)) = 119899119899119899119899119899prime119862119862119899119899) + 6

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119899119899prime119899119862119862119899119899) en fromeorem 10

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 =

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899mod 4)

119899119899 + 12

if 119899119899 119899 1 3 119899mod 4)

119899119899 + 22

if 119899119899 119899 2 119899mod 4)

(45)

119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 (where 1 le 119895119895 119895 1198951198991198991198954119895) for 119899119899 1198990 119899mod4) 119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 (where 1 le 119895119895 le 1198951198991198991198954119895) for119899119899 119899 1 119899mod 4) 119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 2 119899mod4) or 119899119899 119899 3 119899mod4) are 119899119899-sets of 119899119899prime119899119862119862119899119899)en119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) = 6

erefore

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 le 119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 + 6119899 (46)

Now we claim that if 119898119898119899119899119899prime119899119862119862119899119899)) ne 6 for any dominating set 119899119899other than119863119863 then

119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 11989911989910076671007667 ge 119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 + 6119899 (47)

Case 1 If 1198991198991 is any dominating set other than 119863119863 and119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198991) = 4 or 5 then 1198781198991198991119878 ge 119899119899119899119899119899prime119899119862119862119899119899)) + 2Consequently 1198781198991198991119878 + 119898119898119899119899119899

prime119899119862119862119899119899) minus 1198991198991) ge 119899119899119899119899119899prime119899119862119862119899119899)) + 6

Case 2 If 1198991198992 is any dominating set other than 119863119863 and 1198991198991 of119899119899prime119899119862119862119899119899) with 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198992) = 3 then 1198781198991198992119878 ge 119899119899119899 minus 1198951198991198991198953119895)Consequently 1198781198991198992119878 + 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198992) ge 119899119899119899 minus 1198951198991198991198953119895) + 3 119899119899119899119899119899119899prime119899119862119862119899119899)) + 6

Case 3 If 1198991198993 is any dominating set other than 119863119863 1198991198991 and1198991198992 of 119899119899prime119899119862119862119899119899) with 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198993) = 2 or 1 then 1198781198991198993119878 ge 119899119899Consequently 1198781198991198993119878 + 119898119898119899119899119899

prime119899119862119862119899119899) minus 1198991198993) ge 119899119899 + 1 119899 119899119899119899119899119899prime119899119862119862119899119899)) + 6

us from (46) and (47) we have119863119863119863119863119899119899119899prime119899119862119862119899119899))

3 Concluding Remarks

e domination and vulnerability of network are two impor-tant concepts for the network security We have studied animportant measure of vulnerability known as dominationintegrity and investigate domination integrity of splittinggraphs of path and cycle e results reported here throwsome light in the direction to nd the domination integrityof larger graph obtained from the given graph

References

[1] C A Barefoot R Entringer and H C Swart ldquoVulnerabilityin Graphs-a Comparative Surveyrdquo Journal of CombinatorialMathematics and Combinatorial Computing vol 1 pp 13ndash221987

[2] C A Barefoot R Entringer and H C Swart ldquoIntegrity of treesand powers of cyclesrdquo Congressus Numerantium vol 58 pp103ndash114 1987

[3] W Goddard andH C Swart ldquoOn the integrity of combinationsof graphsrdquo Journal of CombinatorialMathematics andCombina-torial Computing vol 4 pp 3ndash18 1988

[4] W Goddard and H C Swart ldquoIntegrity in graphs bounds andbasicsrdquo Journal of Combinatorial Mathematics and Combinato-rial Computing vol 7 pp 139ndash151 1990

[5] A Mamut and E Vumar A Note on the Integrity of MiddleGraphs Lecture Notes in Computer Science vol 4381 SpringerNew York NY USA 2007

ISRN Combinatorics 7

[6] P Duumlndar and A Aytaccedil ldquoIntegrity of total graphs via certainparametersrdquoMathematical Notes vol 76 no 5-6 pp 665ndash6722004

[7] R Sundareswaran and V Swaminathan ldquoDomination Integrityin graphsrdquo in Proceedings of International conference on Mathe-matical and Experimental Physics pp 46ndash57 Narosa PublishingHouse 2010

[8] R Sundareswaran and V Swaminathan ldquoDomination integrityof middle graphsrdquo in Algebra Graph eory and eir Applica-tions T T Chelvam S Somasundaram and R Kala Eds pp88ndash92 Narosa Publishing House Delhi India 2010

[9] S Vaidya and N Kothari ldquoSome new results on dominationintegrity of graphsrdquo Open Journal of Discrete Mathematics vol2 pp 96ndash98 2012

[10] D B West Introduction To Graph eory 2e Prentice-HallNew Delhi India 2003

[11] T Haynes S Hedetniemi and P Slater Fundamentals ofDomination in Graphs Marcel Dekker New York NY USA1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 ISRN Combinatorics

T 2

119899119899 119899119899119899119899119899prime119899119862119862119899119899)) 119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) 119899119899119899119899119899prime119899119862119862119899119899)) + 119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) 119899119899 119898119898119899119899119899prime119899119862119862119899119899) minus 119899119899) 119878119899119899119878 + 119898119898119899119899119899prime119899119862119862119899119899) minus 119899119899)

7 4 6 10 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 3 88 4 6 10 1199071199071 1199071199072 1199071199073 1199071199075 1199071199076 1199071199078 3 99 5 6 11 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 3 910 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 3 1011 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 3 1112 6 6 12 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990712 3 1113 7 6 13 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 3 1214 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1315 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 3 1316 8 6 14 1199071199071 1199071199072 1199071199074 1199071199075 1199071199077 1199071199078 11990711990710 11990711990711 11990711990713 11990711990714 11990711990716 3 14

is implies that

10092491009249119899119899210092491009249 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 119899119899210076671007667 ge 119899119899 + 1119899 (43)

Hence from Table 2 and the results (40) to (43) we concludethat

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 = 119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 11989911989910076671007667

= 119899119899 minus 10078121007812119899119899310078281007828 + 3119899

(44)

Hence the result

eorem 12 For all 119899119899 ge 17 119863119863119863119863119899119899119899prime119899119862119862119899119899)) = 119899119899119899119899119899prime119862119862119899119899) + 6

Proof Let 1199071199071 1199071199072hellip 119907119907119899119899 be the vertices of cycle 119862119862119899119899 and1199061199061 1199061199062hellip 119906119906119899119899 be the vertices corresponding to 1199071199071 1199071199072hellip 119907119907119899119899which are added to obtain 119899119899prime119899119862119862119899119899) en fromeorem 10

119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 =

100761810076181007634100763410076341007634100763410076341007634100763410076341007634100763410076341007626100762610076341007634100763410076341007634100763410076341007634100763410076341007634100763410076421007642

1198991198992

if 119899119899 119899 0 119899mod 4)

119899119899 + 12

if 119899119899 119899 1 3 119899mod 4)

119899119899 + 22

if 119899119899 119899 2 119899mod 4)

(45)

119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 (where 1 le 119895119895 119895 1198951198991198991198954119895) for 119899119899 1198990 119899mod4) 119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 (where 1 le 119895119895 le 1198951198991198991198954119895) for119899119899 119899 1 119899mod 4) 119863119863 = 1199071199071 1199071199074119895119895 1199071199074119895119895+1 119907119907119899119899 (where 1 le 119895119895 le 1198951198991198991198954119895)for 119899119899 119899 2 119899mod4) or 119899119899 119899 3 119899mod4) are 119899119899-sets of 119899119899prime119899119862119862119899119899)en119898119898119899119899119899prime119899119862119862119899119899) minus 119863119863) = 6

erefore

11986311986311986311986310076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 le 119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 + 6119899 (46)

Now we claim that if 119898119898119899119899119899prime119899119862119862119899119899)) ne 6 for any dominating set 119899119899other than119863119863 then

119878119899119899119878 + 11989811989810076511007651119899119899prime 1007649100764911986211986211989911989910076651007665 minus 11989911989910076671007667 ge 119899119899 10076511007651119899119899prime 100764910076491198621198621198991198991007665100766510076671007667 + 6119899 (47)

Case 1 If 1198991198991 is any dominating set other than 119863119863 and119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198991) = 4 or 5 then 1198781198991198991119878 ge 119899119899119899119899119899prime119899119862119862119899119899)) + 2Consequently 1198781198991198991119878 + 119898119898119899119899119899

prime119899119862119862119899119899) minus 1198991198991) ge 119899119899119899119899119899prime119899119862119862119899119899)) + 6

Case 2 If 1198991198992 is any dominating set other than 119863119863 and 1198991198991 of119899119899prime119899119862119862119899119899) with 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198992) = 3 then 1198781198991198992119878 ge 119899119899119899 minus 1198951198991198991198953119895)Consequently 1198781198991198992119878 + 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198992) ge 119899119899119899 minus 1198951198991198991198953119895) + 3 119899119899119899119899119899119899prime119899119862119862119899119899)) + 6

Case 3 If 1198991198993 is any dominating set other than 119863119863 1198991198991 and1198991198992 of 119899119899prime119899119862119862119899119899) with 119898119898119899119899119899prime119899119862119862119899119899) minus 1198991198993) = 2 or 1 then 1198781198991198993119878 ge 119899119899Consequently 1198781198991198993119878 + 119898119898119899119899119899

prime119899119862119862119899119899) minus 1198991198993) ge 119899119899 + 1 119899 119899119899119899119899119899prime119899119862119862119899119899)) + 6

us from (46) and (47) we have119863119863119863119863119899119899119899prime119899119862119862119899119899))

3 Concluding Remarks

e domination and vulnerability of network are two impor-tant concepts for the network security We have studied animportant measure of vulnerability known as dominationintegrity and investigate domination integrity of splittinggraphs of path and cycle e results reported here throwsome light in the direction to nd the domination integrityof larger graph obtained from the given graph

References

[1] C A Barefoot R Entringer and H C Swart ldquoVulnerabilityin Graphs-a Comparative Surveyrdquo Journal of CombinatorialMathematics and Combinatorial Computing vol 1 pp 13ndash221987

[2] C A Barefoot R Entringer and H C Swart ldquoIntegrity of treesand powers of cyclesrdquo Congressus Numerantium vol 58 pp103ndash114 1987

[3] W Goddard andH C Swart ldquoOn the integrity of combinationsof graphsrdquo Journal of CombinatorialMathematics andCombina-torial Computing vol 4 pp 3ndash18 1988

[4] W Goddard and H C Swart ldquoIntegrity in graphs bounds andbasicsrdquo Journal of Combinatorial Mathematics and Combinato-rial Computing vol 7 pp 139ndash151 1990

[5] A Mamut and E Vumar A Note on the Integrity of MiddleGraphs Lecture Notes in Computer Science vol 4381 SpringerNew York NY USA 2007

ISRN Combinatorics 7

[6] P Duumlndar and A Aytaccedil ldquoIntegrity of total graphs via certainparametersrdquoMathematical Notes vol 76 no 5-6 pp 665ndash6722004

[7] R Sundareswaran and V Swaminathan ldquoDomination Integrityin graphsrdquo in Proceedings of International conference on Mathe-matical and Experimental Physics pp 46ndash57 Narosa PublishingHouse 2010

[8] R Sundareswaran and V Swaminathan ldquoDomination integrityof middle graphsrdquo in Algebra Graph eory and eir Applica-tions T T Chelvam S Somasundaram and R Kala Eds pp88ndash92 Narosa Publishing House Delhi India 2010

[9] S Vaidya and N Kothari ldquoSome new results on dominationintegrity of graphsrdquo Open Journal of Discrete Mathematics vol2 pp 96ndash98 2012

[10] D B West Introduction To Graph eory 2e Prentice-HallNew Delhi India 2003

[11] T Haynes S Hedetniemi and P Slater Fundamentals ofDomination in Graphs Marcel Dekker New York NY USA1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Combinatorics 7

[6] P Duumlndar and A Aytaccedil ldquoIntegrity of total graphs via certainparametersrdquoMathematical Notes vol 76 no 5-6 pp 665ndash6722004

[7] R Sundareswaran and V Swaminathan ldquoDomination Integrityin graphsrdquo in Proceedings of International conference on Mathe-matical and Experimental Physics pp 46ndash57 Narosa PublishingHouse 2010

[8] R Sundareswaran and V Swaminathan ldquoDomination integrityof middle graphsrdquo in Algebra Graph eory and eir Applica-tions T T Chelvam S Somasundaram and R Kala Eds pp88ndash92 Narosa Publishing House Delhi India 2010

[9] S Vaidya and N Kothari ldquoSome new results on dominationintegrity of graphsrdquo Open Journal of Discrete Mathematics vol2 pp 96ndash98 2012

[10] D B West Introduction To Graph eory 2e Prentice-HallNew Delhi India 2003

[11] T Haynes S Hedetniemi and P Slater Fundamentals ofDomination in Graphs Marcel Dekker New York NY USA1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of