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
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 . 훾푐 [휇 퐾푚,푛 ] = 9. Consider 퐷 = {푤 ∪ 푈푖 : 푖 = 1,2 … 푛 − 2} is the Proposition 3.10 dominating set in 휇 푇푚,푛 and 푉 − 퐷 = {푉푖 : 1 ≤ 푖 ≤ 푛 − 1}.
For the graphs 휇 퐾푚,푛 and 휇 퐾푚 ,푛 with the maximum Since 퐷 is connected dominating set and whose 푡푐 complement is triple connected. Hence, 훾푐 휇 푇푚,푛 = independence number 훽0 휇 퐾푚,푛 and 훽0 휇 퐾푚,푛 , 푛 − 1. 훽 휇 퐾 ≤ 훽 휇 퐾 . 0 푚,푛 0 푚,푛 Proposition 3.15. Theorem 3.11 For the graphs 휇 푇푚,푛 and 휇 푇푚,푛 where 푛 = 1and 푡푐 In a Wheel graph, with 푛 ≥ 3, 훾푐 휇 푊푛 = 3. 푚 ≥ 4, 푡푐 푡푐 Proof 푖 훾푐 휇 푇푚,푛 + 훾푐 [휇 푇푚,푛 ] = 푛 + 1 푡푐 푡푐 A wheel in a graph is a graph formed by connecting a 푖 훾푐 휇 푇푚,푛 .훾푐 [휇 푇푚,푛 ] = 2 푛 − 1 . single vertex to all the vertices of a cycle. The wheel graph Observation 3.16. has n vertices and 2(푛 − 1) edges. Let 푉(푊 ) = {푣 ∪ 푣 ∶ 푛 푖 1 ≤ 푖 ≤ 푛} and 퐸(푊푛 ) = {푣푣푖 ∶ 1 ≤ 푖 ≤ 푛}.By the For the graphs 휇 푇푚,푛 and 휇 푇푚,푛 , 훾≤2(휇(푇푚,푛 )) ≤ Mycielski’s construction, 푉(휇(푊 )) = 푉(푊 ) ∪ {푢푢 ∶ 푡푐 푡푐 푛 푛 푖 훾(휇(푇푚,푛 )) ≤ 훾푐 [휇 푇푚,푛 ≤ 훾(휇(푇푚,푛 )) ≤ 훾푐 (휇(푇푚,푛 )) 1 ≤ 푖 ≤ 푛} ∪ {푤}.In 휇(푊푛 ), each 푢is adjacent with each Theorem 3.17. vertex of 푁푊푛 (푣), and 푤 is adjacent with each vertex of 푡푐 {푢푢푖 ∶ 1 ≤ 푖 ≤ 푛}. For the Fan graph 퐹1,푛 where 푛 ≥ 4, then훾푐 휇 퐹1,푛 = By the definition of Mycielskian, 푣 is adjacent with each 3. of {푣 ∶ 1 ≤ 푖 ≤ 푛} ∪ {푢 ∶ 1 ≤ 푖 ≤ 푛}. Assume that 푖 푖 Proof 퐷 = {푤, 푢, 푣푖 } is a blast dominating set in 휇 푊푛 .Suppose if The graph 퐹 has 2n + 3 vertices. By the construction 푛 = 4 then 푉(휇(푊4 )) = 푉(푊4) ∪ {푢푢푖 ∶ 1 ≤ 푖 ≤ 4} ∪ 푛+1 {푤}. of Mycielski’s graph,
Choose the minimal blast dominating sets 퐷1 = 푉 휇 퐹1,푛 = 푉 퐹1,푛 ∪ {푢 ∪ 푢푖 : 1 ≤ 푖 ≤ 푛)} ∪ {푤}and 푤, 푢, 푣 , 퐷 = 푤, 푢, 푣 ,퐷 = 푤, 푢, 푣 , 퐷 = 푤, 푢, 푣 1 2 2 3 3 4 4 퐸 휇 퐹1,푛 = 퐸 퐹1,푛 ∪ {푢푢푖 푣: where 푣휖 푁퐹푛 +1 푣푖 ,1 ≤ 푖 which are connected and their complements 푉 − 퐷1 = ≤ 푛)} 푣 ∪ 푣2,푣3, 푣4 ∪ 푢푖 : 1 ≤ 푖 ≤ 4 , 푉 − 퐷2 = 푣 ∪ (푣1,푣3,푣4)∪푢푖:1≤푖≤4,푉−퐷3=푣∪푣1,푣2,푣4∪푢푖:1≤푖≤4,푉− Choose 퐷 = {푤 ∪ 푢푖 ∪ 푣: 1 ≤ 푖 ≤ 푛} as the dominating
퐷4 = 푣 ∪ (푣1, 푣2,푣3) ∪ 푢푖 : 1 ≤ 푖 ≤ 4 are triple set of휇 퐹1,푛 and 푉 − 퐷 = {푣푖 : 1 ≤ 푖 ≤ 푛}. Here 퐷 is a connected. Also 퐷1 , 퐷2 , 퐷3 , 퐷4 = 3. Therefore, the connected dominating set and its complement푉 − 퐷 lies on 푡푐 assumption result holds. Hence 훾푐 휇 푊푛 = 3. the path. Therefore, 퐷 forms a blast dominating set. Hence, 푡푐 훾푐 휇 퐹1,푛 = 3. Proposition 3.12 Result 3.18. Let G be the Mycielski’s graph of Wheel graph such that For the graphs 휇 퐹1,푛 and 휇 퐹1,푛 where 푛 ≥ 3, 휇 푊 and 휇 푊 have no isolated vertices and of order 푛 푛 푖 훾 휇 퐹 + 훾[ 휇 퐹 ] = 4 2푛 + 1, 1,푛 1,푛
푡푐 푡푐 푖푖 훾 휇 퐹1,푛 . 훾[휇 퐹1,푛 ] = 4. 푖 훾푐 휇 푊푛 + 훾푐 [휇 푊푛 ] = 6and 푡푐 푡푐 푡푐 푡푐 푖푖 훾푐 휇 푊푛 .훾푐 [휇 푊푛 ] = 9. 푖푖푖 훾푐 휇 퐹1,푛 + 훾푐 [휇 퐹1,푛 ] = 6 Proof By the above theorem 3.11.the result is true.
Retrieval Number: F11830986S319/2019©BEIESP Published By: Blue Eyes Intelligence Engineering DOI: 10.35940/ijeat.F1183.0986S319 1113 & Sciences Publication
BLAST DOMINATION FOR MYCIELSKI’S GRAPH OFGRAPHS
푖푣 훾푡푐 휇 퐹 . 훾푡푐 [휇 퐹 ] = 9. For theMycielski’s graph of Wheel graphs, 휇 푊푛 and 푐 1,푛 푐 1,푛 휇 푊푛 푛 ≥ 3), Theorem 3.19. 푡푐 (푖)훾≤2(휇 푊푛 ) ≤ 훾(휇 푊푛 ) ≤ 훾푐 (휇 푊푛 ). For the Snake graph 푇 where 푛 ≥ 3, 훾푡푐 휇 푇 = 푛. 푡푐 푛 푐 푛 (푖푖)훾≤2(휇 푊푛 ) ≤ 훾(휇 푊푛 ) ≤ 훾푐 (휇 푊푛 ). Proof Proposition 4.6 Let G be a triangular snake graph is a connected graph all For any graph G , 훾≤2(퐺) ≤ 푖≤2(퐺) ≤ 푖(퐺) ≤ 훾(퐺) ≤ of whose blocks are triangles. A triangularsnake graph is a 훽0(퐺) if and only if the graph G is any one of the triangular cactus whose block-cut vertex graph is a path. and following graphs 휇 퐶푛 , 휇 푃푛 , 휇 퐾푛 ,휇 퐾푚,푛 , it is obtained from the path 푃 = {푣1, 푣2, . . . , 푣푛+1} by 휇 푊푛 , 퐹1,푛 ,푇푚,푛 and 푇푛 . joining 푣푖and 푣푖+1to a new vertex 푢1, 푢2, . . . , 푢푛 . By the construction of Mycielski’s graph, Proposition 4.7 푉 휇 푇 = 푉 푇 ∪ {푉 : 1 ≤ 푖 ≤ 푛} ∪ {푢 : 1 ≤ 푖 ≤ 푛 + 푡푐 푛 푛 푖 푖 For any graph G,훾푐≤2(퐺) = 훾≤2 퐺 = 1 if and only if G 1} ∪ {푤}and is the Mycielski’s graph of 퐶푛 or 퐾푛 or 퐾푚,푛 or 푊푛 or 퐹1,푛 퐸 휇 푇 = 퐸 푇 ∪ {푉 ∪ 푢 푈푉: 푁 푈 푉 , 1 ≤ 푖 ≤ 푛} 푛 푛 푖 푖 푇푛 푖 푖 or 푇푚,푛 or 푇푛 . Assume that 퐷 = 푤 ∪ {푉 : 1,2,3 … 푛 − 1} is a connected 푖 Observations 4.8 dominating set in 휇 푇푛 and 푉 − 퐷 = 푈푖:1,2. . 푛 ∪ {푣푖 : 1,2 … 푛 + 1} is triple connected in 휇 푇푛 . Therefore (i) Every dominating set is a distance-2 dominating set if 푡푐 훾푐 휇 푇푛 = 푛 and only if 퐺is the Mycielski’s graph of푃푛 or퐶푛 or 퐾푛 or 퐾 or 푊 or 퐹 or 푇 or 푇 . But the converse is not Result 3.20. 푚 ,푛 푛 1,푛 푚,푛 푛 true.
For the graphs 휇 푇푛 and 휇 푇푛 where 푛 ≥ 3, (ii) Every dominating set is a distance-2 dominating set if
푖 훾 휇 푇푛 + 훾[휇 푇푛 ] ≤ 푛 + 2 and only if 퐺 is the Mycielski’s graph of푃푛 or퐶푛 or 퐾푛 or 2 푖푖 훾 휇 푇푛 . 훾[휇 푇푛 ] ≤ (푛 + 2) . 퐾푚 ,푛 or 푊푛 or 퐹1,푛 or 푇푚,푛 or 푇푛 . 푡푐 푡푐 푖푖푖 훾푐 휇 푇푛 + 훾푐 [휇 푇푛 ] = 푛 + 2 (iii) Everydistance-2 dominating set is a blast dominating 푡푐 푡푐 푖푣 훾푐 휇 푇푛 . 훾푐 [휇 푇푛 ] = 2푛. set if and only if 퐺is a Mycielski’s graph of푃푛 or퐶푛 or 퐾푛 or 퐾푚 ,푛 or 푊푛 or or 퐹1,푛 or 푇푚,푛 or 푇푛 . IV. RELATIONSHIP WITH OTHER DOMINATION (iv) Everydistance-2 dominating set is a blast dominating PARAMETERS set if and only if 퐺 is a Mycielski’s graph of푃푛 or퐶푛 or 퐾푛 or In this section, some results and bounds related to 퐾푚 ,푛 or 푊푛 or 퐹1,푛 or 푇푚,푛 or 푇푛 . distance-2 domination and independent distance-2 4.9.Exact values and bounds of some standard graphs domination of mycielski’s graphs are discussed. Also exact (i) If any graph G = CPkis the cocktail party graph of k values of some special graphs areobtained. 푡푐 푡푐 vertices where k = 2n for all ≥ 2 , then 훾푐 퐺 = 훾푐≤2 퐺 = Proposition 4.1. 1. (ii) For the n- Andrasfai graph of k vertices where k = 3n- Let 휇 퐶푛 be a graph for 푛 ≥ 3,훾≤2 휇 퐶푛 ≤ 1 for all 푛 ≥ 1, 푖≤2 휇 퐶푛 . 푡푐 푡푐 훾푐 퐴푘 = 훾푐≤2 퐴푘 = 1. 푡푐 Proof (iii) For any Harary graph Hk,n, 훾푐 퐻푘,푛 = 푡푐 Every independence distance-2 dominating set of 휇 퐶푛 is 훾푐≤2 퐻푘,푛 = 1 = 1 for 푘 ≥ 3. a distance-2 dominating set of 휇 퐶푛 . Thus, 훾≤2 휇 퐶푛 ≤ V. BLAST DOMINATION NUMBER FOR THE 푖≤2 휇 퐶푛 . MYCIELSKI’S GRAPH OF ZERO DIVISOR Proposition 4.2. GRAPHS& RESULTS
For the graphs 퐺 and 퐺, 푛 ≥ 3, 훾≤2(퐺) ≤ 훾(퐺) ≤ 훾(퐺) ≤ This section deals with blast domination of Mycielski’s 푡푐 푡푐 훾푐 (퐺) ≤ 훾푐 (퐺) if G is any one of the graphs 휇 퐶푛 or graph of zero divisor graph and its complement graph and 휇 퐾푛 . the bounds for these graphs are attained.The concept zero Proposition 4.3 divisor graphs are introduced by I.Beck in 1988 and further 푡푐 studied by D.D.Anderson and M.Naseer For any graph G,훾푐≤2 퐺 = 훾≤2 퐺 = 1 if and only if G is Mycielski’s graph of푃푛 or 퐶푛 or 퐾푛 or 퐾푚,푛 or 푊푛 or 퐹1,푛 Example 5.1 or 푇푚,푛 or 푇푛 . 2,3,4 6 Proposition 4.4 For theMycielski’s graph of Complete-bipartite graphs
휇 퐾푚,푛 and휇 퐾푚,푛 ,(푚, 푛 ≥ 2) 푡푐 푖 훾 휇 퐾푚,푛 = 훾푐 (휇 퐾푚,푛 ). 푡푐 (푖푖)훾(휇 퐾푚,푛 ) ≤ 훾푐 (휇 퐾푚,푛 ). Proposition 4.5
Published By: Retrieval Number: F11830986S319/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijeat.F1183.0986S319 1114 & Sciences Publication
International Journal of Engineering and Advanced Technology (IJEAT) ISSN: 2249 – 8958, Volume-8, Issue-6S3, September 2019
푡푐 푡푐 푖푣 훾푐 [휇( ]. 훾푐 휇( = 9.
Proposition 5.6 2 If G ((Z n )) and푛 = 푝 where 푝 is an odd prime number,then the blast domination number of G is given by, 푡푐 훾푐 [휇 ((Z n ))] 3. Proposition 5.7 For and푛 = 푝푞where = 5, 푞 = 7 ,then the blast domination number of G is given by, 푡푐 훾푐 [휇 ((Z n ))] 3. Proposition 5.8
For the graphs ((Zn )) and ((Zn )) ,
훾≤2[ ((Z n ) )] ≤ 훾[ )]
[((Z ))] tc[((Z ))] tc [ )] n c n c Observation 5.8 Figure 5.1TheMycielskian graph of 6 For the graphs 퐺 = 휇( (Z )) and 퐺 = 휇( (Z )) , n n Proposition 5.2 훾≤2[휇 (Z n )) = 푖≤2[휇 = 훾≤2 휇( (Z n )) If G ( ) is the Mycielski’s graph of star 2 p = 푖 휇( (Z )) zerodivisor graph, then the blast domination number of G is ≤2 n given by, 훾푡푐 [휇 ((Z ))] 3. 푐 2 p Proposition 5.9
Result 5.3 If the graphs 퐺 = 휇( and 퐺 = 휇( are the
If the graphs 퐺 = 휇( (Z ))and 퐺 = 휇( (Z )) are Mycielski’s graphs of the star zero divisor graph then 2 p 2 p 푖 훾 [휇 + 훾 휇( = 2 the Mycielski’s graphs of star zero divisor graph where 푝 is ≤2 ≤2 a prime number then
푖푖 훾≤2[휇 (Z n )) . 훾≤2 휇( = 1 푖 훾[휇 + 훾 휇( = 4 푖푖푖 푖 [휇 + 푖 휇( = 2 ≤2 ≤2 푖푖 훾[휇 . 훾 휇( = 4 푖푣 푖≤2[휇 . 푖≤2 휇( = 1 푡푐 푡푐 푖푖푖 훾푐 [휇 + 훾푐 휇( = 5 Proof
푖푣 훾푡푐 [휇 . 훾푡푐 휇( = 6. 푐 푐 By the above observation 5.8 the result is true. Proposition 5.4 Obsevation 5.10 For G ((Z )) , if푛 = 3푝 where 푝 is a prime (i) Every distance-2 dominating set is a blast distance-2 n number, the blast domination number of G is given by, dominating set in 휇( and [휇( ] .Converse is 푡푐 also true. 훾푐 [휇 ((Z3 p ))] 3. (ii) Every independent distance-2 dominating set is a blast
Result 5.5 distance-2 dominating set in μ( and [휇( ].
Converse is also true. If the graphs 퐺 = 휇( (Z3 p )) and 퐺 = 휇( (Z3 p )) are the Mycielski’s of graphs zero divisor graph where 푝 is a Proof prime number (푝 > 3) then
The results follows from 푖 훾≤2[휇 푖 훾[휇( (Z )) ] + 훾 휇( = 5 3 p tc [((Z ))] and c2 n 푖푖 훾[휇( ]. 훾 휇( = 6
푡푐 푡푐 푖푖푖 훾푐 [휇( ] + 훾푐 휇( = 6
Retrieval Number: F11830986S319/2019©BEIESP Published By: Blue Eyes Intelligence Engineering DOI: 10.35940/ijeat.F1183.0986S319 1115 & Sciences Publication
BLAST DOMINATION FOR MYCIELSKI’S GRAPH OFGRAPHS
tc 푖푖 푖≤2[휇 (Z n )) c2[((Zn ))]
VI. APPLICATIONS OF BLAST AND BLAST DISTANCE - 2 DOMINATING SETS We extended the concept of dominating sets to Blast and Blast distance - 2 dominating sets, there are more useful models to many real- world problems. Indeed, much of the motivation for the study of Blast and Blast distance – 2 domination arises from problems involving locating optimally a hospital, police station, fire station, or any other emergency service facility.
VII.CONCLUSION In this paper, we defined the notion of blast and blast distance-2 domination forMycielski’s graph of 퐶푛 ,퐾푛 ,퐾푚,푛 and 푊푛 graphs. We attained many bounds on these two new parameters. Also Applications of Blast and Blast distance - 2 dominating sets have been discussed. It would be interesting to determine the results for general graphs.
REFERENCES 1. BondyJ.A.and U.S.R. Murty, Graph theory, (Springer, Berlin, 2008). 2. ChartrandG and P. Zhang, Introduction to graph theory, (McGraw-Hill Inc., New York, 2005). 3. HaynesT.W.,HedetniemiS. T and SlaterP.J., Fundamentals of Domination in Graphs, Marcel Dekker, Inc.,New York,1998. 4. Kulli.V.R.,(2012). Advances in domination theory, Vishwa International Publications, Gulbarga, India. Yorozu, M. Hirano, K. 5. Lin W, J. Wu, P.C.B. Lam and G. Gu, Several parameters of generalized Mycielskians,Discrete Appl. Math., 154(8)(2006), 1173{1182}. 6. G. Mahadevan, A. Ahila and S. Avadayappan,Blast domination number of a graph, Middle –East J. Sci. Res.,25(5) (2017), 977- 981International Journal of Pure and Applied Mathemati 7. Mahadevan .G,. Ahila . A and SelvamAvadayappan,Blast domination number for ϑ- Obrazom, International Journal of Pure and Applied Mathematics, Volume 118 No. 7 2018, 111-117 8. Mahadevan .G,.Ahila . A and SelvamAvadayappan, Blast Domination Number on square and cube graphs,Contemp. Stud. Discrete Math., Vol. 1, No. 1, 2017, pp. 31-35 9. J.Paulraj Joseph, M.K.AngelJebitha, P.Chithra Devi and G.Sudhana, Triple connected graphs, Indian Journal of Mathematics and Mathematical Sciences, Vol.8, No.I(2012),61–75. 10. Tian.F and Xu J.M., A note on distance domination number of graphs, Australasian J. Combin.43(2009), 181-190.
Published By: Retrieval Number: F11830986S319/2019©BEIESP Blue Eyes Intelligence Engineering DOI: 10.35940/ijeat.F1183.0986S319 1116 & Sciences Publication