For a graph $G=(V,E)$, a sequence $S=(v_1, v_2, \cdots, v_k)$ of distinct vertices of $G$ is called \emph{dominating sequence} if $N_G[v_i]\setminus \bigcup_{j=1}^{i-1}N[v_j]\neq\varnothing$ and is called \emph{total dominating sequence} if $N_G(v_i)\setminus \bigcup_{j=1}^{i-1}N(v_j)\neq\varnothing$ for each $2\leq i\leq k$. The maximum length of (total) dominating sequence is denoted by ($\gamma_{gr}^t)\gamma_{gr}(G)$. In this paper we compute (total) dominating sequence numbers for generalized corona products of graphs

Maimani, Hamid Reza

Moosavi Majd, Seyedeh Maryam

English

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NIŠ)Ser. Math. Inform. Vol. 35, No 4 (2020), 1231–1237https://doi.org/10.22190/FUMI2004231MGRUNDY DOMINATION SEQUENCES IN GENERALIZEDCORONA PRODUCTS OF GRAPHSSeyedeh Maryam Moosavi Majd and Hamid Reza Maimani∗ ∗c© 2020 by University of Nǐs, Serbia | Creative Commons Licence: CC BY-NC-NDAbstract. For a graph G = (V,E), a sequence S = (v1, . . . , vk) of distinct vertices of Git is called a dominating sequence if NG[vi] \⋃i−1j=1 N [vj ] 6= ∅ and is called a total dom-inating sequence if NG(vi) \⋃i−1j=1N(vj) 6= Ø for each 2 ≤ i ≤ k. The maximum lengthof (total) dominating sequence is denoted by (γtgr) γgr(G). In this paper we compute(total) dominating sequence numbers for generalized corona products of graphs.Keywords: dominating sequence; total dominating sequence; generalized corona prod-ucts.1. IntroductionIn this paper, G is a simple graph with the vertex set V = V (G) and theedge set E = E(G). For notation and graph theoretical terminology, we generallyfollow [8]. The order |V | and the size |E| of G is denoted by n = n(G) andm = m(G), respectively. For every vertex v ∈ V , the open neighborhood NG(v) ofv is the set {u ∈ V (G) : uv ∈ E(G)} and the closed neighborhood of v is the setNG[v] = NG(v) ∪ {v}. The degree of a vertex v ∈ V is degG(v) = dG(v) = |NG(v)|.Theminimum degree and themaximum degree of a graphG are denoted by δ = δ(G)and ∆ = ∆(G), respectively. We write Pn for the path of order n, Cn for the cycleof order n, and Kn for the complete graph of order n. A subset D of V (G) is calleda dominating set of G if every vertex of G is either in D or adjacent to at leastone vertex in D. The domination number of G, denoted by γ(G), is the numberof vertices in a smallest dominating set of G. A total dominating set of G is aset D of vertices of G such that every vertex is adjacent to a vertex in D. Thetotal domination number of G, denoted by γt(G), is the minimum cardinality of atotal dominating set. A dominating set of cardinality γ(G) (γt(G)) is called a γ-setReceived July 12, 2020; accepted August 21, 20202020 Mathematics Subject Classification. Primary 05C69; Secondary 05C76∗Corresponding Author12311232 S.M. Moosavi Majd and H.R. Maimani(γt-set). For further information about various domination sets in graphs, we referreader to [9, 10].Let G be a graph of order n and letH1, H2, . . . , Hn, be n graphs. The generalizedcorona product, is the graph obtained by taking one copy of graphsG,H1, H2, · · · , Hnand joining the ith vertex of G to every vertex of Hi. This product is denoted byG◦∧ni=1Hi. If each Hi is isomorphic to a graph H , then generalized corona productis called the corona product of G and H and is denoted by G ◦H .Let G be a graph of size m and H be a graph. The edge corona product, denotedby G ⋄H , is the graph obtained by taking one copy of G and m copies of H , andthen joining two end-vertices of the ith edge ei of G to every vertex of ith copy ofH . The neighborhood corona, denoted by G ⋆H , is the graph obtained by taking ncopies of H and for each i, 1 ≤ i ≤ n, the ith copy of H being adjacent to verticesof NG[vi]. It is not difficult to see that G⋄H is the same as G◦∧ni=1Hi, where eachHi is a disjoint union of deg(vi) copies of H and G ⋆ H is the same as G ◦ ∧ni=1Hi,where each Hi is a disjoint union of deg(vi) + 1 copies of H .Based on the domination number and the total domination number, variousGrundy domination invariants have been introduced in recent years by some authors[1, 5, 6] and then they continued the study of these concepts in [3, 2, 4, 7].In [5] the first type of Grundy dominating sequence was introduced. Let S =(v1, . . . , vk) be a sequence of distinct vertices of a graph G. The correspondingset {v1, . . . , vk} of vertices from the sequence S will be denoted by Ŝ. A sequenceS = (v1, . . . , vk) is called a closed neighborhood sequence if, for each i,NG[vi] \i−1⋃j=1NG[vj ] 6= Ø.If for a closed neighborhood sequence S, the set Ŝ is a dominating set of G, then Sis called a dominating sequence of G. Clearly, if S = (v1, v2, . . . , vk) is a dominatingsequence for G, then k ≥ γ(G). We call the maximum length of a dominating se-quence in G the Grundy domination number of G and denote it by γgr(G). The cor-responding sequence is called a Grundy dominating sequence of G or γgr-sequenceof G.Total dominating sequences were introduced in [6], when G is a graph withoutisolated vertices. Using the same notation as in the previous paragraph, we saythat a sequence S = (v1, . . . , vk) is called an open neighborhood sequence if, for each2 ≤ i ≤ k,NG(vi) \i−1⋃j=1NG(vj) 6= Ø.Any open neighborhood sequence S, where Ŝ is a total dominating set is called atotal dominating sequence. The maximum length of a total dominating sequencein G is called the Grundy total domination number of G and denoted by γtgr(G).Grundy Domination Sequences 1233The corresponding sequence is called a Grundy total dominating sequence of G ora γtgr-total sequence.An additional variant of the Grundy (total) domination number was introducedin [1]. Let G be a graph without isolated vertices. A sequence S = (v1, . . . , vk),where vi ∈ V (G), is called a Z − sequence if for each i,NG(vi) \i−1⋃j=1NG[vj ] 6= Ø.Then the Z-Grundy domination number γZgr(G) of the graph G is the length of alongest Z-sequence.Let S1 = (v1, . . . , vn) and S2 = (u1, . . . , um), n,m ≥ 1, be two sequences inG, with Ŝ1 ∩ Ŝ2 = Ø. The concatenation of S1 and S2 is defined as the sequenceS1 ⊕ S2 = (v1, . . . , vn, u1, . . . , um). Clearly ⊕ is an associative operation on the setof all sequences, but is not commutative. If S2 = {v}, then S1 ⊕ S2 is denoted byS1 ⊕ v.In the next section, we compute Grundy domination numbers for generalizedcorona products of graphs and based on, we find Grundy domination numbers ofedge and neighborhood corona products of graphs.2. Main ResultsIn this section we give the exact value of (total) Grundy domination numbers forgeneralized corona products, and compute them for corona product of some specialgraphs. First we state two necessary known propositions.Proposition 2.1. [6] For n ≥ 4 even, γtgr(Pn) = n and γtgr(Cn) = n − 2, whilefor n ≥ 3 odd, γtgr(Pn) = γtgr(Cn) = n− 1.Proposition 2.2. [5, 1] For n ≥ 3, γgr(Cn) = γZgr(Cn) = n− 2, while for n ≥ 2,γgr(Pn) = γZgr(Pn) = n− 1.we are now state and proof the our first main result.Theorem 2.1. Let G and H1, H2, . . . , Hn be n+1 graphs without isolated vertices.Thenγgr(G ◦ ∧ni=1Hi) =n∑i=1γgr(Hi) + γZgr(G).Proof. Set K = G ◦ ∧ni=1Hi. Let S = (v1, . . . , vk) be a Z-Grundy sequence of Gand Si be a γgr-sequence of Hi for 1 ≤ i ≤ n. It is not difficult to see thatS1 ⊕ v1 ⊕ S2 ⊕ v2 ⊕ . . .⊕ Sk ⊕ vk ⊕ Sk+1 ⊕ Sk+2 ⊕ . . .⊕ Sn1234 S.M. Moosavi Majd and H.R. Maimaniis a dominating sequence for K. This implies that γgr(K) ≥∑ni=1 γgr(Hi)+γZgr(G).Let T be a γgr-sequence of K such that |T̂⋂V (G)| is minimum among allγgr-sequences. Suppose that T̂⋂V (G) = {v1, . . . , vt}, where (v1, . . . , vt) is a subse-quence of T . If t > γZgr(G), then (v1, . . . , vt) is not a Z-sequence forG and thus, thereexists 1 ≤ l ≤ t such that NG(vl)\⋃l−1i=1 NG[vi] = ∅. But NK [vl]\⋃l−1i=1 NK [vi] 6= ∅,since (v1, . . . , vt) is a sub-sequence of T . If T̂⋂V (Hl) 6= ∅, then there exists an ele-ment z ∈ V (Hl) such that one of the (v1, . . . , vl, z) or (v1, . . . , vi−1, z, vi, . . . , vl) is asubsequence of T . If (v1, . . . , vl, z) is a subsequence of T , then NK [z]\⋃li=1 NK [vi] =Ø, which is a contradiction. Hence (v1, . . . , vi−1, z, vi, . . . , vl) is a subsequence ofT . Therefore there exists x ∈ NK [vl] \⋃l−1i=1 NK [vi]⋃NK [z]. Since vl ∈ NK [z] andNG(vl) \⋃l−1i=1 NG[vi] = Ø, we conclude that x 6= vl. In addition, x ∈ V (Hl) andx, z are not adjacent vertices, and x /∈ T̂ . Now, by replacing vl by x in T , we obtaina γgr-sequence T′, such that |T̂ ′⋂V (G)| < |T̂⋂V (G)|, which is a contradiction.Hence T̂⋂V (Hl) = Ø. Again consider a vertex x ∈ V (Hl) and put x instead ofvl in T . Then we obtain a γgr-sequence T′ such that the size of intersection ofT̂ ′ and V (G) is less than the size of intersection of T̂ and V (G). This is a con-tradiction and so we conclude that |T̂⋂V (G)| ≤ γZgr(G). It is not difficult to see|T̂⋂V (Hi)| ≤ γgr(Hi) for 1 ≤ i ≤ n and thus γgr(K) ≤∑ni=1 γgr(Hi)+γZgr(G).The following corollary is an easy consequence of Theorem 2.1 and Proposition2.2.Corollary 2.1. For n,m ≥ 3γgr(Cn ◦ Cm) = n(m− 1)− 2, γgr(Pn ◦ Pm) = mn− 1,γgr(Cn ◦ Pm) = nm− 2, γgr(Pn ◦ Cm) = n(m− 1)− 1.we are now stat and proof our second main result.Theorem 2.2. Let G andH1, H2, . . . , Hn be graphs without isolated vertices. Thenγtgr(G ◦ ∧ni=1Hi) =n∑i=1γtgr(Hi) + γZgr(G).Proof. Consider the sequenceT = S1 ⊕ v1 ⊕ S2 ⊕ v2 ⊕ . . .⊕ Sk ⊕ vk ⊕ Sk+1 ⊕ Sk+2 ⊕ . . .⊕ Sn,where S = (v1, . . . , vk) is a Z-Grundy sequence of G and Si’s are γtgr-sequences ofHi’s for 1 ≤ i ≤ n. We show that T is a γtgr-sequence for K = G ◦ ∧ni=1Hi. Letx ∈ T̂ . Hence there exists either 1 ≤ i ≤ n such that x ∈ Ŝi or 1 ≤ j ≤ k for whichx = vj . If x = vj , then there exists y ∈ NG(vj) \⋃j−1t=1 NG[vt]. Hence y 6= vt forGrundy Domination Sequences 12351 ≤ t ≤ j− 1 and therefore y ∈ NK(vj)\⋃j−1t=1 NK [vt]⋃(⋃jt=1 Nk[St]). This impliesthatNK(vj) \j−1⋃t=1NK [vt]⋃(j⋃t=1NK [St]) 6= Ø.The same argument can be apply when x ∈ Ŝi. Since clearly T̂ is a total dominatingset, we conclude that T is a total dominating sequence of G. Henceγtgr(K) ≥n∑i=1γtgr(Hi) + γZgr(G).Now suppose that T is a γtgr-sequence of K such that |T̂⋂V (G)| is minimumamong all γtgr-sequences of G. Suppose that T̂⋂V (G) = {v1, . . . , vt} and t >γZgr(G). Hence (v1, . . . , vt) is not a Z-sequence for G. Therefore, there exists 1 ≤l ≤ t such that NG(vl)\⋃l−1i=1 NG[vi] = Ø. If T̂⋂V (Hl) = Ø, then by replacing vl byx ∈ V (Hl), we can construct a γtgr-sequence T′ such that |T̂ ′⋂V (G)| < |T̂⋂V (G)|,which is a contradiction. Hence T̂⋂V (Hl) 6= Ø. If there exists x ∈ T̂⋂V (Hl)such that x appears after vl in the sequence T , then (vl, x) is a subsequence ofT and NK(x) \ NK(vl) 6= Ø. Since NG(vl) \⋃l−1i=1 NG[vi] = Ø, we conclude thatNK(x) \ NK(vl) = {vl} and hence T̂⋂V (Hl) = {x}. Now choose y ∈ N(x) andreplace vl by y in T . Again we obtain a γtgr-sequence T′ such that |T̂ ′⋂V (G)| <|T̂⋂V (G)|, which is a contradiction. Hence all elements of T̂⋂V (Hl) appearbefore vl in the sequence T . Hence there exists y ∈ V (Hl) such that y ∈ NK(vl) \⋃x∈T̂⋂V (Hl)NK(x). Since degHl(y) ≥ 1, there exists z ∈ V (Hl) which is adjacentto y. Clearly z /∈ T̂ and by changing vl with z, we get a γtgr-sequence T′ suchthat |T̂ ′⋂V (G)| < |T̂⋂V (G)|, which is a contradiction. This argument impliesthat |T̂⋂V (G)| ≤ γZgr(G). One can easily check that |T̂⋂V (Hi)| ≤ γtgr(Hi) for1 ≤ i ≤ n and so we conclude that γtgr(K) ≤∑ni=1 γtgr(Hi) + γZgr(G).Corollary 2.2. Let G be a graph of order n and size m and H be a graph withoutisolated vertices. Then γgr(G ⋄H) = 2mγgr(H) + γZgr(G) and γtgr(G ⋆ H) = (2m+n)γtgr(H) + γZgr(G).Proof. Note that G ⋄H is the same as G ◦ ∧ni=1Hi, where Hi is the disjoint unionof deg(vi) copies of H . Hence by Theorem 2.1,γgr(G ⋄H) = γgr(G ◦ ∧ni=1Hi) =n∑i=1γgr(Hi) + γZgr(G) = 2mγgr(H) + γZgr(G).The proof of the second part of the corollary is similar.Corollary 2.3. Let G be a connected graph of order n. Then γtgr(G ◦K1) = 2n.1236 S.M. Moosavi Majd and H.R. MaimaniProof. Suppose that V (G) = {v1, . . . , vn} is the vertex set of G. It is not difficult tosee that sequence (u1, u2, . . . , un, v1, v2, . . . , vn), where ui is the vertex of K1, whichis adjacent to vi, is a Grundy total domination sequence of G ◦K1.Corollary 2.4. Let G be a nontrivial connected graph of order n. Then γtgr(G ◦H) = nγtgr(H) + γZgr(G), for any nontrivial connected graph H.As a similar argument to proof of Theorem 2.1, we can find the Z-Grundy domina-tion number of corona product of graphs.Theorem 2.3. Let G and H1, H2, . . . , Hn be n+1 graphs without isolated vertices.ThenγZgr(G ◦ ∧ni=1Hi) =n∑i=1γZgr(Hi) + γZgr(G).REFERENCES1. B. Brešar, Cs. Bujtas, T. Gologranc, S. Klavzar, G. Kosmrlj, B. Patkos, Z. Tuza and M.Vizer, Grundy dominating sequences and zero forcing sets, Discrete Optim. 26 (2017),66–77.2. B. Brešar, C. Bujtas, T. Gologranc, S. Klavzar, G. Kosmrlj, B. Patkos, Z. Tuza, M.Vizer, Dominating sequences in grid-like and toroidal graphs Electron. J. Combin., 23(2016), P4.34 (19 pages).3. B. Brešar, T. Gologranc and T. Kos, Dominating sequences under atomic changes withapplications in Sierpinski and interval graphs, Appl. Anal. Discrete Math. 10 (2016),518–531.4. B. Brešar, Kos and Terros, Grundy domination and zero forcing in Kneser graphs, ArsMath. Contemp., 17(2019), 419-430.5. B. Brešar, T. Gologranc, M. Milanič, D. F. Rall, R. Rizzi, Dominating sequences ingraphs. Discrete Math. 336 (2014), 22-36.6. B. Brešar, M. A. Henning, D. F. Rall, Total dominating sequences in graphs. DiscreteMath. 339 (2016) 1165-1676.7. B. Brešar, T. Kos, G. Nasini, P. Torres, Total dominating sequences in trees, splitgraphs, and under modular decomposition, Discrete Optim., 28(2018), 16-30.8. G. Chartrand, L. Lesniak, Graphs and digraphs, Third Edition, CRC Press,(1996).9. T. W. Haynes, S. Hedetniemi, P. Slater, Fundamentals of Domination in Graphs, CRCPress, (1998).10. M. A. Henning and A. Yeo, Total domination in graphs, (Springer Monographs inMathematics.) ISBN-13: 987-1461465249 (2013).Seyedeh Maryam Moosavi MajdDepartment of MathematicsScience and Research Branch, Islamic Azad UniversityTehran, Iranmoosavi.majd@gmail.comGrundy Domination Sequences 1237Hamid Reza MaimaniMathematics Section, Department of Basic SciencesShahid Rajaee Teacher Training UniversityP.O. Box 16785-163Tehran, Iranmaimani@ipm.ir

GRUNDY DOMINATION SEQUENCES IN GENERALIZED CORONA PRODUCTS OF GRAPHS

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GRUNDY DOMINATION SEQUENCES IN GENERALIZED CORONA PRODUCTS OF GRAPHS

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