Domination Number of Square of Cartesian Products of Cycles

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1 Ope Joural of Discrete Matheatics, 01,, Published Olie October 01 i SciRes Doiatio Nuber of Square of artesia Products of ycles Morteza Alishahi, Sakieh Hoseii Shalaee Islaic Azad Uiversity, Nazarabad Brach Eail: ortezaalishahi@gailco Received August 01; accepted 6 October 01; published 9 October 01 opyright 01 by authors ad Scietific Research Publishig Ic This work is licesed uder the reative oos Attributio Iteratioal Licese ( BY Abstract A set S V ( G is a doiatig set of G if every vertex of V ( G S is adjacet to at least oe vertex of S The cardiality of the sallest doiatig set of G is called the doiatio uber of G The square G of a graph G is obtaied fro G by addig ew edges betwee every two vertices havig distace i G I this paper we study the doiatio uber of square of graphs, fid a boud for doiatio uber of square of artesia product of cycles, ad fid the exact value for soe of the Keywords Doiatio Nuber, Square of a Graph, artesia Product 1 Itroductio The usual graph theory otios ot herei, refer to [1] The eighborhood of vertex u is deoted by N u v V G : uv E G N u = N u u ( = { ( ( } ad the close eighborhood of vertex u is deoted by [ ] ( { } Let S V ( G, the eighborhood ad closed eighborhood of S are defied as N( S N( u { } = ad N[ S] = N[ u] If u V ( G, the N ( ( 1 (, u S k u = v V G d uv k If S V ( G ad u V ( G the d( u, S = i { d( u, v v S} The diaeter of G deoted by ( dia( G = ax, ( dg ( u, v uv V G A set S V ( G is a doiatig set of G if every vertex of V ( G S adjacet to at least oe vertex of S The cardiality of the sallest doiatig set of G, deoted by ( G called the doiatio uber of G A doiatig set of cardiality ( G u S, dia G is defied as is, is is called a -set of G [] A doiatig set S is a iial doiatig set if o proper subset S S is a doiatig set Give ay graph G, its square graph V G ad two vertices are adjacet wheever they are at G is a graph with vertex set ( How to cite this paper: Alishahi, M ad Shalaee, SH (01 Doiatio Nuber of Square of artesia Products of ycles Ope Joural of Discrete Matheatics,,

2 M Alishahi, S H Shalaee distace 1 or i G For exaple K S V G is a -distace doiatig set of G if dg ( us, = 1 or for every vertex of V ( G S The cardiality of the sallest -distace doiatig set of G, deoted by ( G, is called -distace doiatio uber of G Every -distace doiatig set of G is a doiatig set of G = G The artesia product of Graphs G ad H deoted by G H is a G, so ( ( = A set ( graph with vertex set V ( G H = V ( G V ( H ad the edge set E( G H = (( u, v,( z, w : ( uz E( G & v = w or ( u = z& vw E( H { } The graph G H is obtaied by locatig copies H i of grpah H istead of vertices of G ad coectig the correspodig vertices of H i to H j if vertex v i is adjacet to v j i G G H is isoorphic to H G We deote a cycle with vertices by ad a path with vertices by P The bipartite geraph K 1,3 is aed claw Preliiaries Results Theore 1 Let G be a graph The uv E G G uv G a If (, the ( ( c b If uv E ( G, the ( G + uv ( G Proof a Every doiatig set of G uv is a doiatig set of G so ( G uv ( G b Every doiatig set of G is a doiatig set of G uv G + uv G Theore [3] A doiatig set S is a iial doiatig set if ad oly if for each vertex u S, oe of the followig coditios holds: a u is a isolated vertex of S b there exist a vertex v V ( G S for which N( v S = { u} Theore 3 [3] If G is a graph with o isolated vertices ad S is a iial doiatig set of G, the V ( G S is a doiatig set of G V G S is a doiatig + so ( ( Proof Let S be a -set of G S is a iial doiatig set of G By Theore 3, ( set of G too, so S V ( G S, so S Theore 4 [4] If G is a coected claw free graph, the ( G 3 Theore [] Let G be a graph The ( G ( G 1+ ( G Sice ( = ( P =, by Theores 4 ad we have the followig corollary orollary 6 ( = ( P = 3 Vizig cojecture G H G H [6] Let G ad H be two graphs The ( ( ( 3 Doiatio Nuber of Square of Graphs Theore 7 Let S be a doiatig set of G The S is a iial doiatig set of vertex u S satisfies at least oe of the followig coditios: v V G S N v S = u a There exists a vertex ( for which ( { } b d( uw, > for every vertex w S { u} Proof If u S ad u does t satisfy coditios a ad b, the the set S { u} G if ad oly if each is a doiatig set of G that is cotradictio oversely, let S be a doiatig set of G but ot iial The there exists a vertex u S such that S { u} is a doiatig set of G, too So d( vs, { u} = 1 or for every v V ( G S ; therefore S does t satisfy i coditio a I additio d( us, { u} = 1 or, so S does t satisfy i coditio b 89

3 M Alishahi, S H Shalaee dia G, the ( G δ ( G Theore 8 If ( 3 Proof Let d( u = δ ( G Sice dia( G 3, the set ( ( G N ( u d ( u δ ( G = = ( ( v N u dia G, the ( ( ( N u is a doiatig set of G Therefore Theore 9 If ( 4 G d v d u, for every u V ( G Proof Let u be a arbitrary vertex of G Let S( u = N( N( u { u} Sice dia( G 4, (, ( every v V ( G Therefore S( u is a doiatig set of G S( u ( ( d( v 1 v N u ad N( u d( u Hece ( S ( ( ( ( 1 = ( ( ( ( v N u v N u Theore 10 Let G be a graph The (( G K = ( G Proof Let V ( G { u u u } H H H K i G K the vertices u 1, u,, u =,,, ad 1,,, be the copies of 1 d vs u, for =, correspodig to Let S = { u, u,, u } be a -set of G The the set S V ( G H t1 t tk cotais a vertex of each copies K 1, K,, K Theore 11 For every 3 Proof The graphs 3 ad 4 3 ad is a -set of ( t t tk, ( Let = uu 1 uu 1, = ( that G K Sice S = S, the result holds 4 are coplete graphs, therefore ( 3 ( 4 1 = = So the result holds for Sice ( = 4, by the Theore we have ( the other had by Figure 1 the set : 0,1,, 1 S = uk k = is a doiatig set of size So (, therefore ( = Theore 1 For every 1, ( P = Proof ( P ( P ( P ( P O for = = = =, ad the result holds for these graphs Let P = uu 1 u, Sice ( P = 4, by Theore we have ( P By Figure the set: uk + 3 : k = 0,1,, 1 if 0 ( od S = uk+ 3 : k = 0,1,, 1 u if 1,, 3 od uk+ 3 : k = 0,1,, 1 { u 1} if 4 ( od { } ( is a doiatig set of size for P, so ( P ; therefore ( P = Theore For every, 1, (( P P, ad for every, 3, (( Proof The graphs P P ad have vertices ad every vertex u doiates at least vertices P P ad ( (Figure 3, so the result holds i ( ( P P ( By Theore, ( or ( we ca cover all the vertices of P P or equals the iiu uber of diaods like Figure 4 90

4 M Alishahi, S H Shalaee Figure 1 A doiatig set of Figure A doiatig set of P Figure 3 Doiated vertices by u i ( P P ad ( Figure 4 Doiated vertices by oe vertex i ( P P ad ( I this paper we use short display or sd to show the graphs P P ad for siplicity; it eas that we do t draw the edges of these graphs ad draw oly their vertices Theore 14 For every, 1 ( k t kt ( ( = kt, ( Proof By Theore we have ( I Figure that is sd of It is deteried by a -set of size for ( Therefore ( ; hece ( We ca obtai sd of k t Figure i k rows ad t colus Hece ( ( = with doiatig set of size kt for ( ( k t by locatig kt copies of k t kt By Theore we have 91

5 M Alishahi, S H Shalaee (( k t kt, so ( k t ( 3 Figure A doiatig set of size for ( ( = kt Theore 1 ( = 3, for every 3 Proof Sice 3 = K3, by Theore 10 ad orollary 6 we have (( 3 = (( K3 = ( = 3 ( 4 Theore 16 ( 4 =, = 4,,6,7, ad k if t = 0, k if t = 1, ( k if t =,3, 4, (( 4 t + ( 4 4 Proof By Theore we have ( I Figure 6 it is deteried by a doiatig set of 4 size for ( 4 4, = 4,,6,7, so for these graphs we have (( 4 = I Figure 6, the seveth colu of sd of 4 7 (fro left to right is siilar to the first colu of sd of 4 P1, 4 P ad 4, = 3, 4,,6,7 By settig sd of k graphs 4 7 ad oe sd of 4 P or 4 3 or 4 4 or 4 cosecutively fro left to right such that the first colu of every sd of graph locates o the last colu of sd of the previous graph, we ca obtai a sd of 4 + t with a doiatig set of size ( k for ( 4 + t, t =,3, 4, By the sae settig for sd of k graphs 4 7 we ca obtai a sd of 4 with a doiatig set of size k for ( 4 Also by the sae settig for sd of k 1 graphs 4 7 ad oe sd of 4 6 we ca obtai a sd of 4 with a doiatig set of size k for ( 4 Theore 17 (( =, = 3, 4, 6, ad (( t + ( Proof By Theore we have ( 3k if t = 0, 3k if t = 1, 3k+ if t =,3, 4, 3( k if t = I Figure 7 it is deteried by a doiatig set for 9

6 M Alishahi, S H Shalaee Figure 6 A doiatig set for (, = 1,,, 7 4 Figure 7 A doiatig set for (, = 1,,, 7 93

7 M Alishahi, S H Shalaee ( P, ( P ad ( 1, = 3, 4,,6,7 ( By Figure 7 we have (, = 3, 4, 6 So for these graphs equality holds I Figure 7, the seveth colu of sd of 7 (fro left to right is siilar to the first colu of sd of P1, P ad, = 3, 4,,6,7 By settig sd of k graphs 7 ad oe sd of P or 3 or 4 cosecutively fro left to right such that the first colu of every sd of graph locates o the last colu of the previous sd of graph, we ca obtai a sd of + t with a doiatig set of size 3k + for ( + t, t =,3, 4 By the sae settig for sd of k graphs 7 we ca obtai a sd of with a doiatig set of size 3k for ( ad by the sae settig for sd of k graphs 7 ad oe sd of we ca obtai a sd of + with a doiatig set of size 3( k for ( + Also by the sae settig for sd of k 1 graphs 7 ad oe sd of 6 we ca obtai a sd of with a doiatig set of size 3k for ( Refereces [1] West, DB (001 Itroductio to Graph Theory d Editio, Pretice-Hall, Upper Saddle River [] Hayes, T, Hedetiei, S ad Slater, PJ (1997 Fudaetals of Doiatio i Graphs M dekker, Ic, New York [3] Ore, O (196 Theory of Graphs Aerica Matheatical Society olloquiu Publicatios, 38 (Aerica Matheatical Society, Providece, RI [4] ockaye, EJ, Ko, W ad Shepherd, FB (198 Iequalities ocerig Doiatig Sets i Graphs Techical Report DM-370-IR, Departet of Matheatics, Uiversity of Victoria [] Walikar, HB, Acharya, BD ad Sapathkuar, E (1979 Recet Developets i the Theory of Doiatio i Graphs I MRI Lecture Notes i Math Mehta Research Istitute of Matheatics, Allahabad, Vol 1 [6] Vizig, VG (1963 The artesia Product of Graphs Vycisl Sistey, 9,