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International J.Math. Combin. Vol.3(2017), 72-80 Strong Domination Number of Some Cycle Related Graphs Samir K. Vaidya (DepartmentofMathematics,SaurashtraUniversity,Rajkot-360005,Gujarat,India) Raksha N. Mehta (AtmiyaInstituteofTechnologyandScience,Rajkot-360005,Gujarat,India) E-mail: [email protected],[email protected] Abstract: Let G = (V(G),E(G)) be a graph and u,v V(G). If uv E(G) and ∈ ∈ deg(u) deg(v),thenwesaythatustronglydominatesvorvweaklydominatesu. Asubset ≥ D of V(G) is called a strong dominating set of G if every vertex v V(G) D is strongly ∈ − dominated by some u D. The smallest cardinality of strong dominating set is called ∈ a strong domination number. In this paper we explore the concept of strong domination numberand investigate strong domination numberof some cycle related graphs. Key Words: Dominating strong set, Smarandachely strong dominating set, strong domination number,d balanced graph. − AMS(2010): 05C69, 05C76. §1. Introduction Inthis paperweconsiderfinite, undirected, connectedandsimple graphG. Thevertexsetand edge set of the graph G is denoted by V(G) and E(G) respectively. For any graph theoretic terminology andnotations we rely upon Chartrandand Lesniak [2]. We denote the degree of a vertexv inagraphGbydeg(v). ThemaximumandminimumdegreeofthegraphGisdenoted by (G) and δ(G) respectively. △ A subset D V(G) is independent if no two vertices in D are adjacent. A set D V(G) ⊆ ⊆ of vertices in the graph G is called a dominating set if every vertex v V(G) is either an ∈ element of D or is adjacent to an element of D. A dominating set D is a minimal dominating ′ set if no proper subset D D is a dominating set. The domination number γ(G) of G is the ⊂ minimum cardinality of a minimal dominating set of the graph G. A detailed bibliography on the concept of domination can be found in Hedetniemi and Laskar [7] as well as Cockayne and Hedetniemi [3]. A dominating set D V(G) is called an independent dominating set if it is ⊆ also an independent set. The minimum cardinality of an independent dominating set in G is called the independent domination number i(G) of the graph G. For the better understanding of domination and its related concepts we refer to Haynes et al [6]. 1ReceivedOctober 04,2016, AcceptedAugust21,2017. StrongDominationNumberofSomeCycleRelatedGraphs 73 We will give some definitions which are useful for the present work. Definition 1.1([10]) For graph G and uv E(G), we say u strongly dominates v (v weakly ∈ dominates u ) if deg(u) deg(v). ≥ Definition 1.2([10]) A subset D is a strong(weak) dominating set sd set(wd set) if ev- − − ery vertex v V(G) D is strongly(weakly) dominated by some u in D. The strong(weak) ∈ − domination number γ (G)(γ (G)) is the minimum cardinality of a sd set(wd set). st w − − Generally, for a subset O V(G) with O isomorphic to a special graph, for instance ⊂ h iG a tree, a subset D of V(G) is a Smarandachely strong(weak) dominating set of G on O if S every vertex v V(G) D O is strongly(weakly) dominated by some vertex in D . Clearly, S ∈ − − if O= , D is nothing else but the strong dominating set of G. S ∅ The concepts of strong and weak domination were introduced by Sampathkumar and Pushpa Latha [10]. In the same paper they have defined the following concepts. Definition 1.3 The independent strong(weak) domination number i (G) (i (G)) of the graph st w G is the minimum cardinality of a strongly(weakly) dominating set which is independent set. Definition 1.4 Let G=(V(G),E(G)) be a graph and D V(G). Then D is s-full (w-full) if ⊂ every u D strongly (weakly) dominates some v V(G) D. ∈ ∈ − Definition 1.5 A graph G is domination balanced (d-balanced) if there exists an sd-set D and 1 a wd-set D such that D D =φ. 2 1 2 ∩ Several results on the concepts of strong and weak domination have also been explored by Domke et al [4]. The bounds on strong domination number and the influence of special vertices on strong domination is discussed by Rautenbach [8,9] while Hattingh and Henning have investigatedbounds on strongdomination number of connectedgraphs in [5]. For regular graphs γ = γ = γ as reported by Swaminathan and Thangaraju in [11]. Therefore we st w consider the graph G which is not regular. §2. Main Results We begin with propositions which are useful for further results. Proposition 2.1([10]) For a graph G of order n, γ γ n (G). st ≤ ≤ −△ Proposition 2.2([1]) For a nontrivial path P , n n if n 1(mod 3) γst(Pn)=⌈n3⌉ and γw(Pn)= ln3m+1 otherw≡ise  3 l m Proposition 2.3([1]) For cycle C , γ (C )=γ (C )=γ(C )= n . n st n w n n ⌈3⌉ Proposition2.4([11]) Foranynon regulargraph G, γ (G)+ (G)=nand γ (G)+δ(G)=n st w △ 74 SamirK.VaidyaandRakshaN.Mehta if and only if (1) for every vertex u of degree δ, V(G) N[u] is an independent set and every vertex in − N(u) is adjacent to every vertex in V(G) N(u). − (2)foreveryvertexvofdegree ,V(G) N[v]isindependent,eachvertexinV(G) N(v)is △ − − ofdegree δ+1and novertex ofN(v)stronglydominates twoormorevertices of V(G) N[v]. ≥ − Proposition 2.5([10]) For a graph G, the following statements are equivalent. (1) G is d-balanced; (3) There exists an sd-set D which is s-full; (3) There exists an wd-set D which is w-full. Theorem 2.6 Let G be the graph of order n. If there exists a vertex u with deg(u )= and 1 1 △ deg(u )=m, where 2 i n then, γ (G)=γ(G). i st ≤ ≤ Proof Let G be the graph of order n and let u be the vertex with deg(u )= (G). The 1 1 △ set V(G) N(u ) contains the vertices of degree m. It is clear that the graph G contains two 1 − types ofvertices: a vertexofdegree andremainingverticesofdegreem. The vertexu γ 1 st △ ∈ - set as it is of maximum degree. To prove the result we consider following two cases. Case 1. N[u ]=V(G). 1 If N[u ] = V(G) implies that γ(G) = 1. Hence deg(D) > deg(V(G) D). Therefore 1 − u D strongly dominates V(G) D. Thus γ (G)=γ(G)=1. 1 st ∈ − Case 2. N[u ]=V(G). 1 6 Let us partition the vertex set V(G) into V and V . Now to construct a dominating 1 2 set or a strong dominating set of minimum cardinality the vertex u must belong to every 1 strong dominating set. So let N[u ] V and remaining n 1 vertices are in V . Now 1 1 2 ∈ −△− the vertices in V are of degree m. Thus the vertices V forms a regular graph. For regular 2 2 graphs γ(G) = γ (G) = γ (G). Let k be the domination number of vertex set V . Therefore st w 2 γ(G)=γ (G)=γ (V )+γ (V )=1+k. st st 1 st 2 In any case, if G contains a vertex of degree (G) and remaining vertices of same degree △ m then γ (G)=γ(G). st 2 Corollary 2.7 γ(K )=γ (K )=i (K )=1. 1,n st 1,n st 1,n Corollary 2.8 γ(W )=γ (W )=i (W )=1. n st n st n Definition 2.9 One point union C(k) of k copies of cycle C is the graph obtained by taking n n v as a common vertex such that any two cycles C(i) and C(j) (i =j) are edge disjoint and do n n 6 not have any vertex in common except v. Corollary 2.10 γ (C(k))=γ(C(k))=1+k n 3 , for n 3. st n n ⌈ −3 ⌉ ≥ Proof Let vp,vp,...vp be the vertices of pth copy of cycle C for 1 p k, k N and 1 2 n n ≤ ≤ ∈ StrongDominationNumberofSomeCycleRelatedGraphs 75 v be the common vertex in graph Ck such that v = v1 = v2 = v3 = = vp. Consequently n 1 1 1 ··· 1 V(Ck) =kn k+1. | n | − The deg(v)=2k which is of maximum degree, then it must be in every dominating set D and the vertex v will dominate 2k+1 vertices. Now to dominate the remaining k disconnected copies of path each of length n 3 we − require minimum k n 3 vertices. −3 This implies that γ(Ck) 1+k n 3 . Let us partition the vertex set V(Ck) into V (Ck) (cid:6) (cid:7)n ≥ −3 n 1 n and V (Ck) such that V(Ck) = V (Ck) V (Ck) depending on the degree of vertices. Let 2 n n 1(cid:6) n (cid:7) 2 n V (Ck)containN[v]whichformsastargraphK . Thus,fromaboveCorollary2.7γ(K )= 1 n S 1,2k 1,n 1. LetV (Ck)containtheremainingvertices,thatis, V (Ck) = V(Ck) - V (Ck) =kn k+1 2 n | 2 n | | n | | 1 n | − (2k+1) = kn 3k in k copies. Thus, in one copy there are n 3 vertices which forms a − − − path of order n 3. Therefore, from above Proposition 2.2, γ(P )= n 3 . For k copies of − n−3 −3 path the domination number is γ[k(P )] = k n 3 . Hence, γ(Ck) = γ(K ) + γ[k(P )] n−3 −3 n (cid:6) 1(cid:7),n n−3 = 1+k n 3 , for n 3. Therefore D is a dominating set of minimum cardinality. Thus, D is −3 ≥ (cid:6) (cid:7) also the strong dominating set of minimum cardinality. Therefore, (cid:6) (cid:7) n 3 γ (C(k))=γ(C(k))=1+k − , st n n ⌈ 3 ⌉ for n 3. ≥ 2 Definition 2.11 Duplication of a vertex v by a new edge e =uv in a graph G results into a i ′ ′ ′ graph G such that N(u)= v ,v and N(v )= v ,u . ′ ′ i ′ ′ i ′ { } { } Theorem 2.12 If G is the graph obtained by duplication of each vertex of graph G by a new ′ edge then γ (G)=γ(G)=n. st ′ ′ Proof Let V(G) be the set of vertices and E(G) be the set of edges for the graph G. Let us denote vertices of graph G by u ,u ,u ,u . Hence V(G) = n and E(G) = m. 1 2 3 n ··· | | | | Each vertex of G is duplicated by a new edge. Let us denote these new added vertices by v ,v ,v ,v and w ,w ,w , w respectively. Hence, the obtained graph G contains 1 2 3 n 1 2 3 n ′ ··· ··· 3n vertices and 3n+m edges. Thus the degree of u (1 i n) will increase by two and i ≤ ≤ the degree of v and w (1 i n) is two. The graph G contains n vertex disjoint cycles of i i ′ ≤ ≤ order 3. By Proposition 2.3, γ (C ) = 1. Thus minimum n vertices are essential to strongly st 3 dominate n vertex disjoint cycles. Hence, γ (G) n. Since u are the vertices of maximum st ′ i ≥ degree, they must be in every strong dominating set. We claim that it is enough to take u i in strong dominating set as the vertices v and w are adjacent to a common vertex u . Thus, i i i D = u ,u ,u ,u is the only strong dominating set with minimum cardinality. Hence, 1 2 3 n { ··· } γ (G)=γ(G)=n. st ′ ′ 2 Theorem 2.13 If G is the graph obtained by duplication of each vertex of graph G by a new ′ edge then G is d balanced. ′ − Proof As argued in Theorem 2.12,D = u ,u ,u ,u is the only strong dominating 1 2 3 n { ··· } 76 SamirK.VaidyaandRakshaN.Mehta set. Henceitisthestrongdominatingsetwithminimumcardinality. Theverticesu (1 i n) i ≤ ≤ strongly dominates v and w in V(G) D where (1 i n). Thus, D is s full. Hence from i i ′ − ≤ ≤ − Proposition 2.5 G is d balanced. ′ − 2 Definition 2.14 The switching of a vertex v of G means removing all the edges incident to v and adding edges joining to every vertex which is not adjacent to v in G. We denote the resultant graph by G. Theorem 2.15 IfeC is the graph obtained by switching of an arbitrary vertex v in cycle C , n n (n>3) then, f 1 if n=4 γst(Cn)= 2 if n=5  3 if n 6 f ≥  Proof Let v ,v ,v ...,v be the vertices of the cycle C . Without loss of generality we 1 2 3 n n switch the vertex v of C . We consider following cases to prove the theorem. 1 n Case 1. n=4. The graph C is obtained by switching of vertex v in cycle C which is same as K . 4 1 4 1,3 Hence D = v is the only strong dominating set as discussed in Corollary 2.7. It is the only 3 { } f strong dominating set with minimum cardinality. Therefore the strong domination number γ (C )=1. st 4 Casfe 2. n=5. The graph C obtained by switching of vertex v in cycle C . The degree deg(v ) = 2, 5 1 5 1 deg(v ) = deg(v ) = 1 while deg(v ) = deg(v ) = 3. The vertex v strongly dominates v , v 2 5 3 4 3 1 2 f and v along with itself. It is enough to take the vertex v in the strong dominating set to 4 4 strongly dominate the vertex v . Thus D = v ,v is the only strong dominating set with 5 3 4 { } minimumcardinality. HencearbitraryswitchingofavertexofcycleC resultsintoγ (C )=2. 5 st 5 Case 3. n 6. f ≥ LetC be the graphobtainedby switchingofvertexv incycle C . The degreedeg(v )= n 1 n 1 n 3 while deg(v ) = deg(v ) = 1 and remaining n 3 vertices are of degree three. Thus, 2 n − f − V(C ) =n. Bythe Proposition2.1γ (C ) n (C )=n (n 3),implying γ (C ) 3. n st n n s n | | ≤ −△ − − ≤ Thedegreedeg(v )=n 3whichisofmaximumdegree,thatis,v mustbeineverystrong 1 1 f − f f f dominating set andv will stronglydominaten 2 verticesexceptthe pendant verticesv and 1 2 − v . Henceeitherthesependantverticesmustbeineverystrongdominatingsetorthesupporting n vertices v and v . Thus, D = v ,v ,v or D = v ,v ,v or D = v ,v ,v or n 1 3 1 1 2 n 2 1 3 n 1 3 1 2 n 1 − { } { − } { − } D = v ,v ,v are strong dominating sets with minimum cardinality. Therefore, 4 1 n 3 { } γ (C )=3. st n for n 6. f ≥ 2 StrongDominationNumberofSomeCycleRelatedGraphs 77 Illustration 2.16 In Figure 2.1, the solid vertices are the elements of strong dominating sets of C as shown below. 6 v 1 f v 6 v 2 v v 5 3 v 4 γ(C )=γ (C )=3 6 st 6 Figure 2.1 f f Corollary 2.17 γ (C )=γ(C ) for n>3. st n n Proof WecontinfuewithtfheterminologyandnotationsusedinTheorem2.15andconsider the following cases to prove the corollary. Case 1. n=4. As shown in Theorem 2.15, D = v is the only strong dominating set with minimum 3 { } cardinality which is also the dominating set of minimum cardinality. As discussed in Corollary 2.7, γ (C )=γ(C ). st 4 4 Case 2. n=5. f f As shown in Theorem 2.15, D = v ,v is the only strong dominating set with minimum 3 4 { } cardinality which is also the dominating set of minimum cardinality. Hence γ (C )=γ(C ). st 5 5 Case 3. n 6. ≥ f f As shown in Theorem 2.15 we have obtained four possible strong dominating sets. The strong dominating sets D = v ,v ,v or D = v ,v ,v or D = v ,v ,v or 1 1 2 n 2 1 3 n 1 3 1 2 n 1 { } { − } { − } D = v ,v ,v are strong dominating sets with minimum cardinality which are also the 4 1 n 3 { } dominating set of minimum cardinality. Thus, γ (C )=γ(C ) , for n 6. st n n ≥ 2 Theorem 2.18 If C is the graph obtained by swiftching offan arbitrary vertex v in cycle C n n then,C (n>3) is d balanced. n − f Pfroof WecontinuewiththeterminologyandnotationsusedinTheorem2.15andconsider the following cases to prove the corollary. Case 1. n=4. As discussed in Theorem 2.15 the set D = v is the strong dominating with minimum 3 { } cardinality. ThesetD iss fullsincethevertexv stronglydominatesremainingthreevertices 3 − 78 SamirK.VaidyaandRakshaN.Mehta in V(C ) D. Hence from Proposition 2.5 C is d balanced. 4 4 − − Case 2. n=5. g f As shownin Theorem2.15,the setD = v ,v is a strongdominating setwith minimum 3 4 { } cardinality. The set D = v ,v is s full since v (i=3,4) strongly dominates v , v and v 3 4 i 2 4 5 { } − in V(C ) D. Hence from Proposition 2.5 C is d balanced. 5 5 − − Case 3. n 6. f ≥ f In Theorem 2.15 we have obtained the strong dominating set D = v ,v ,v of mini- 2 1 3 n 1 { − } mumcardinality. ThesetD iss fullasv ,v andv stronglydominatesremainingvertices 2 1 3 n 1 − − in V(C ) D . Thus from Proposition 2.5 C (n 6) is d balanced. n 2 n − ≥ − 2 Definfition 2.19 The book B is a graph Sf P where S =K . m m 2 m 1,m × Theorem 2.20 γ (B )=2 for m 3. st m ≥ Proof LetS bethegraphwithverticesu,u ,u ,u ,u whereuisthevertexofdegree m 1 2 3 m ··· m and u ,u ,u ,u are pendant vertices. Let P be the path with vertices a and a . We 1 2 3 m 2 1 2 ··· consider v = (u,a ),v = (u ,a ),v = (u ,a ) ,v = (u ,a ) and w = (u,a ),w = 1 1 1 1 2 2 1 m m 1 2 1 ··· (u ,a ),w =(u ,a ) ,w =(u ,a ). Hence V(B ) =2m+2. 1 2 2 2 2 m m 2 m ··· | | In B there is no vertex with degree 2m+1, implying that γ(B ) > 1. The deg(v) = m m deg(w) = m+1 are the vertices of maximum degree. Let us partition the vertex set V(B ) m into V and V such that V(B ) = V V . Let N[v] V and N[w] V . Then in both the 1 2 m 1 2 1 2 ∈ ∈ partitionsastargraphK isformed. ThusfromaboveCorollary2.7,γ(K )=γ (K )= 1,m S 1,m st 1,m 1. Thus,it is enoughto takev and w in strongdominating setas it stronglydominates 2m+2 vertices. Therefore D = v,w is the strong dominating set with minimum cardinality. Hence, { } γ (B )=2 if m 3. st m ≥ 2 Illustration 2.21 In Figure 2.2, the solid vertices are the elements of strong dominating set of B as shown below. 3 v w 1 1 v w v2 w2 v3 w3 γ(B )=γ (B )=2 3 st 3 Figure 2.2 Corollary 2.22 γ (B )=γ(B ) for m 3. st m m ≥ StrongDominationNumberofSomeCycleRelatedGraphs 79 Proof AsshowninTheorem2.20wehaveobtainedthestrongdominatingsetD = v,w . { } ThesetD alsoformsthe dominatingsetofminimum cardinality. Thus,γ (B )=γ(B ),for st m m n 3. ≥ 2 Theorem 2.23 The book graph B is d balanced. m − Proof In Theorem 2.20 we have obtained the strong dominating set D = v,w of min- { } imum cardinality. The vertex v strongly dominates v ,v ,v while the vertex w strongly 1 2 m ··· dominatew ,w ,w inV(B ) D respectively. HenceD iss fullset. HencefromPropo- 1 2 m m ··· − − sition 2.5 the book graph B is d balanced. m − 2 §3. Concluding Remarks The strong domination in graph is a variant of domination. The strong domination number of variousgraphsareknown. We haveinvestigatedthe strongdominationnumber ofsomegraphs obtained from C by means of some graph operations. This work can be applied to rearrange n the existing security networkin the caseof high alertsituationand to beef up the surveillance. Acknowledgment The authors are highly thankful to the anonymous referees for their critical comments and fruitful suggestions for this paper. References [1] R. BoutrigandM. Chellali,A note on arelationbetweenthe weakand strongdomination numbers of a graph, Opuscula Mathematica, Vol. 32, (2012), 235-238. [2] G.ChartrandandL.Lesniak,GraphandDigraphs,4thEd.,ChapmanandHall/CRCPress (2005). [3] E.J.CockayneandS.T.Hedetniemi,Towardsatheoryofdominationingraphs,Networks 7, (1977), 247-261. [4] G. S. Domke, J. H. Hattingh, L. R. Markus and E. Ungerer, On parameters related to strong and weak domination in graphs, Discrete Mathematics, Vol. 258, (2002), 1-11. [5] J. H. Hattingh, M. A. Henning, On strong domination in graphs, J. Combin. Math. Combin. Comput., Vol 26, (1998), 33-42. [6] T.W.Haynes,S.T.HedetniemiandP.J.Slater,Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998. [7] S. T. Hedetniemi and R. C. Laskar,Bibliographyon dominationin graphs and some basic definitions of domination parameters,Discrete Math., Vol. 86, (1990), 257-277. [8] D. Rautenbach, Bounds on the strong domination number, Discrete Mathematics, Vol. 215, (2000), 201-212. 80 SamirK.VaidyaandRakshaN.Mehta [9] D. Rautenbach, The influence of special vertices on strong domination, Discrete Mathe- matics, Vol. 197/198,(1999), 683-690. [10] E.SampathkumarandL.PushpaLatha,Strongweakdominationanddominationbalance in a graph, Discrete Mathematics, Vol. 161, (1996), 235-242. [11] V. Swaminathan and P. Thangaraju, Strong and weak domination in graphs, Electronic Notes in Discrete Mathematics, vol. 15, (2003), 213-215.