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Blast Domination for Mycielski's Graph of Graphs International Journal of Engineering and Advanced Technology (IJEAT) ISSN: 2249 – 8958, Volume-8, Issue-6S3, September 2019 Blast Domination for Mycielski’s Graph of Graphs K. Ameenal Bibi, P.Rajakumari AbstractThe hub of this article is a search on the behavior of is the minimum cardinality of a distance-2 dominating set in the Blast domination and Blast distance-2 domination for 퐺. Mycielski’s graph of some particular graphs and zero divisor graphs. Definition 2.3[7] A non-empty subset 퐷 of 푉 of a connected graph 퐺 is Key Words:Blast domination number, Blast distance-2 called a Blast dominating set, if 퐷 is aconnected dominating domination number, Mycielski’sgraph. set and the induced sub graph < 푉 − 퐷 >is triple connected. The minimum cardinality taken over all such Blast I. INTRODUCTION dominating sets is called the Blast domination number of The concept of triple connected graphs was introduced by 퐺and is denoted by tc . Paulraj Joseph et.al [9]. A graph is said to be triple c (G) connected if any three vertices lie on a path in G. In [6] the Definition2.4 authors introduced triple connected domination number of a graph. A subset D of V of a nontrivial graph G is said to A non-empty subset 퐷 of vertices in a graph 퐺 is a blast betriple connected dominating set, if D is a dominating set distance-2 dominating set if every vertex in 푉 − 퐷 is within and <D> is triple connected. The minimum cardinality taken distance-2 of atleast one vertex in 퐷. The blast distance-2 푡푐 over all triple connected dominating sets is called the triple domination number 훾푐≤2(퐺) is the minimum cardinality of connected domination number of G and is denoted by a blast distance-2 dominating set in 퐺. 훾푡푐 (퐺). Also Mahadevan et.al introduced the concept of Definition 2.5 complementary triple connected domination number of a A distance -2dominating set D V of a graph G is an graph and A.Ahilaet.al.,introducedBlast Domination Number of a graph with a real life application. independentdistance -2dominating set if the induced sub graph <D> has no edges. The independent distance - II. THE MYCIELSKI CONSTRUCTION [5] 2domination number iG2 ()is the minimum cardinality of a The open neighborhood of a vertex 푣 in a graph G minimal independentdistance -2dominating set. The minimal independent distance -2 dominating set in a denoted by 푁퐺(푣)isthe set of all vertices of 퐺, which are graph G is an independent distance -2 dominating set that adjacent to 푣. Also, 푁퐺[푣] = 푁퐺(푣) ∪ {푣}is called the closed neighborhood of v in the graph G. In this paper, by G contains no independent distance - 2 dominating set as a one means a connected graph. proper subset. From a graph G, by Mycielski’s construction one can get The distance -2 open neighborhood of a vertex 푣 ∈ 푉 is a graph μ (G) with 푉 (휇 (퐺)) = 푉 ∪ 푈 ∪ {푤},where the set, 푁≤2(푣) of vertices within distance of two from푣. 푉 = 푉 (퐺) = {푣1,. 푣푛 } , 푈 = {푢1, . , 푢푛 } ,and III. MAIN RESULTS 퐸 (휇 (퐺)) = 퐸 (퐺) ∪ {푢푖 푣 ∶ 푣 ∈ 푁퐺 (푣푖) ∪ {푤} , 푖 = 1, . , 푛} .For each 0 ≤ 푖 ≤ 푛, 푣 and 푢 are called the 푖 푖 Blast domination in the Mycielski’s graph of퐶푛 ,퐾푛 corresponding vertices of μ (G) and denote 퐶 (푣푖 ) = ,푊푛 퐾푚,푛,퐹1,푛 , 푇푚 ,푛 , and 푇푛 graphs 푢 , 퐶 (푢 ) = 푣 . Moreover, for subsets A ⊆U, B ⊆V , one 푖 푖 푖 In this section, blast domination number of some graphs denotes:퐶 (퐴) = {퐶 (푢 ) ∶ 푢 ∈ 퐴}, 퐶 (퐵) = {퐶 (푣 ) ∶ 푖 푖 푖 are investigated. Many bounds for this parameter is 푣 ∈ 퐵}. Also, x ↔ y is denoted, when{x, y} is an edge. 푖 obtained. Definition 2.1[7] Example 3.1 A graph 퐺 is said to be triple connected, if any three vertices of 퐺 lie on a path. Definition 2.2[10] A set퐷 of vertices in a graph 퐺 is a distance-2 dominating set if every vertex in 푉 − 퐷 is within distance-2 of atleast one vertex in 퐷. The distance-2 domination number 훾≤2(퐺) Revised Manuscript Received on 14 August, 2019. K. AmeenalBibi, P.G. and Research Department of Mathematics, D.K.M.College for Women (Autonomous), Vellore, Tamilnadu, India. P.Rajakumari, P.G. and Research Department of Mathematics, D.K.M.College for Women (Autonomous), Vellore, Tamilnadu, India. (Email: [email protected]). Retrieval Number: F11830986S319/2019©BEIESP Published By: Blue Eyes Intelligence Engineering DOI: 10.35940/ijeat.F1183.0986S319 1111 & Sciences Publication BLAST DOMINATION FOR MYCIELSKI’S GRAPH OFGRAPHS Let G be theMycielski’s graph of Cycle graph such that 휇 퐶푛 and 휇 퐶푛 have no isolated vertices and of order 2푛 + 1. 푡푐 푡푐 푖 훾푐 휇 퐶푛 + 훾푐 [휇 퐶푛 ] ≤ 푛 + 2and 푡푐 푡푐 푖푖 훾푐 휇 퐶푛 .훾푐 [휇 퐶푛 ] ≤ 3(푛 − 1). Proposition 3.4 For the graphs 휇 퐶푛 and 휇 퐶푛 with maximum independence number 훽0 휇 퐶푛 and 훽0 휇 퐶푛 (푖)훽0 휇 퐶푛 + 훽0 휇 퐶푛 = 2푛 2 (푖푖)훽0 휇 퐶푛 .훽0 휇 퐶푛 = 푛 . Theorem 3.5 푡푐 In a Complete graph, where푛 ≥ 3, 훾푐 휇 퐾푛 = 3. Proof A complete graph is a simple, connected,undirected graph (a) Mycielskian graph of 푷ퟕ in which every pair of distinct vertices is connected by a unique edge. Let 푉 (퐾푛 ) = {푥푖 ∶ 1 ≤ 푖 ≤ 푛}. By the construction of Mycielski’s graph, 푉 (휇 (퐾푛 )) = 푉 (퐾푛 ) ∪ {푦푖 ∶ 1 ≤ 푖 ≤ 푛} ∪ {푧}and퐸 (휇 (퐾푛 )) = 퐸 (퐾푛 ) ∪ {푦푖 푥 ∶ 푥 ∈ 푁퐾푛 (푥푖 ) ∪ {푧} , 푖 = 1, 2, . 푛} . Since 푧 is adjacent with each vertex of {푦푖 ∶ 1 ≤ 푖 ≤ 푛}, also 휇 (퐾푛 ) contains a n-clique. Assume that 퐷 = {푧, 푦푖 , 푥푖 : 푖 = 1, 2, . 푛} is a blast dominating set in 휇 퐾푛 . If 푛 = 3 Choose and fix any of the corresponding vertices {푧, 푦1, 푥2},{푧, 푦2,푥1},{푧, 푦3, 푥3}, which are connected and its complement {푦2, 푦3, 푥1, 푥3}, {푦1,푦3,푥2, 푥3},{푦1,푦2, 푥1, 푥2} 푡푐 forms a blast dominating set. Therefore, 훾푐 휇 퐾3 = 푡푐 3.Successively, the assumption is true. Hence 훾푐 휇 퐾푛 = (b) Mycielskian graph of C6 3. Figure 1.1 Result 3.6 Theorem 3.2 Let G be a Mycielski’s graph of Complete graph such that 푡푐 In a Cycle graph, for푛 ≥ 4, 훾푐 휇 퐶푛 = 푛 − 1. 휇 퐾푛 and 휇 퐾푛 have no isolated vertices of order 2푛 + 1. 푡푐 푡푐 Proof 푖 훾푐 휇 퐾푛 + 훾푐 [휇 퐾푛 ] = 6and 푡푐 푡푐 A cycle in a graph is a closed walk consists of a sequence 푖푖 훾푐 휇 퐾푛 . 훾푐 [휇 퐾푛 ] = 9. ofvertices starting and ending at the same vertex, with each Remark 3.7 two consecutive vertices in the sequence adjacent to each other in the graph. For the graphs 휇 퐾푛 and 휇 퐾푛 with the maximum Let 푉 (퐶푛 ) = {푥푖 ∶ 1 ≤ 푖 ≤ 푛}be the set of vertices of independence number 훽0 휇 퐾푛 and 훽0 휇 퐾푛 , 퐶푛 taken in cyclic order. By the construction of Mycielski’s (푖)훽0 휇 퐾푛 + 훽0 휇 퐾푛 = 2푛 graph, 2 (푖푖)훽0 휇 퐾푛 .훽0 휇 퐾푛 = 푛 . 푉 (휇 (퐶푛 )) = 푉 (퐶푛 ) ∪ {푦푖 : 1 ≤ 푖 ≤ 푛} ∪ Theorem 3.8 {푧} and퐸 (휇 (퐶푛 )) = 퐸 (퐶푛 ) ∪ {푦푖 (푥) ∶ 푥 ∈ 푁퐶푛 (푥푖 ) ∪ {푧} , 푖 = 1, 2, . , 푛} . In a Complete-bipartite graph 퐾푚,푛 where 푚, 푛 ≥ 푡푐 Assume that 퐷 = {푢, 푢푖 : 푖 = 1, 2, . 푛 − 2} is a blast 2, 훾푐 휇 퐾푚,푛 = 3. dominating set in 퐶 . 푛 Proof If 푛 = 5, fix the vertex set {푢, 푢1, 푢2, 푢3} which is Let 푉 (휇(퐾 )) = {푥 ∶ 1 ≤ 푖 ≤ 푚} [ {푦 ∶ 1 ≤ 푗 ≤ connected and its complement 푉 − 퐷 is the {푣푖 ∪ 푢4, 푢5} 푚,푛 푖 푖 푡푐 which is lie on a path.Therefore 훾푐 휇 퐶푛 = 푛 − 1. 푛} and 푡푐 푚 Successively, hence 훾푐 휇 퐶푛 = 푛 − 1. 퐸 퐾푚,푛 = ∪푖=1 [{푒푖푗 = 푥푖 푦푗 ∶ 1 ≤ 푖 ≤ 푛} . Result 3.3 Published By: Retrieval Number: F11830986S319/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijeat.F1183.0986S319 1112 & Sciences Publication International Journal of Engineering and Advanced Technology (IJEAT) ISSN: 2249 – 8958, Volume-8, Issue-6S3, September 2019 By Mycielski’s construction, Result 3.13 ′ ′ 푉 (휇 (퐾푚 ,푛 )) = 푉 (퐾푚 ,푛 ) [ {푥 푖 ∶ 1 ≤ 푖 ≤ 푚} [ 푦 ∶ 1 푗 For the graphs 휇 푊푛 and 휇 푊푛 with the maximum ≤ 푗 ≤ 푛 푧 ] . independence number 훽0 휇 푊푛 and 훽0 휇 푊푛 , Choose the minimal blast dominating sets 퐷1 = 훽0 휇 푊푛 ≤ 훽0 휇 푊푛 푧, 푦1, 푦푚 +푛 , 퐷2 = 푧, 푦2,푦푚+푛−1 , Theorem 3.14. 퐷푛 = {푧, 푦푖 ,푦푚 }and|퐷1|, |퐷2|. |퐷푛 | = 3 which are For the Tadpole graph 푇푚 ,푛 where 푛 = 1and 푚 ≥ 4, connected and its complement < 푉 − 퐷 > is triple 푡푐 훾푐 휇 푇푚,푛 = 푛 − 1. 푡푐 connected. Hence 훾푐 [휇 퐾푚,푛 ] = 3. Proof Result 3.9 Let 푇푚 ,1 be the Tadpole graph with joining the cycle 퐶푛 Let G be a Mycielski’s graph of Complete-bipartite graph and path 푃푛 with 푚 = 4, 푛 = 1. Let 푉 푇푚,1 = 퐾 such that 휇 퐾 and 휇 퐾 have no isolated 푚,푛 푚,푛 푚,푛 {푉1, 푉2, . 푉푛 }.By the construction of Mycielski’s graph, vertices of order 2푛 + 1, 푉 휇 푇푚,푛 = 푉 푇푚 ,1 ∪ 푈푖 : 1 ≤ 푖 ≤ 푛 ∪ 푤 and 푡푐 푡푐 푖 훾푐 휇 퐾푚,푛 + 훾푐 [휇 퐾푚,푛 ] = 6and 푡푐 푡푐 퐸 휇 푇푚,푛 = 퐸 푇푚 ,1 ∪ {푈푖 푉: 푉 ∈ 푁푇푚 ,푛 (푉푖 ), 푖 = 1,2 … 푛 푖푖 훾푐 휇 퐾푚,푛 .
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