BLAST DOMINATION NUMBER OF A GRAPH FURTHER RESULTS. Gandhigram Rural Institute - Deemed University, Gandhigram, Dindigul

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1 BLAST DOMINATION NUMBER OF A GRAPH FURTHER RESULTS G.Mahadevan 1, A.Ahila 2*, Selvam Avadayappan 3 1 Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram, Dindigul drgmaha2014@gmail.com 2* Dept. of Mathematics, Kalasalingam University, Virudhunagar , Tamilnadu, India dr.aaasundar@gmail.com 3 Dept.of Mathematics, VHNSN College Virudhunagar , Tamilnadu, India selvam_avadayappan@yahoo.co.in ABSTRACT G.Mahadevan et.al., introduced the concept ofblast Domination Number of a graph with a real life application. A subset S of V of a non-trivial graph G is said to be a Blast Dominating Set, if S is a connected dominating set and the induced sub graph V S is triple connected. The minimum cardinality taken over all Blast Dominating sets is called the Blast Domination Number and is denoted by γ c G. As an extension of that paper, in this paper, we investigate some more results on BDN and BDN with 3-regular graphs and with other graph theoretical parameters. Key words:domination number, connected domination number, Triple connected domination number and chromatic number. AMS (2010): 05C69 1. INTRODUCTION By a graph,g(v, E),we mean a finite, undirected graph with neither multiple edges nor loops. The order and size of G are denoted by nandm respectively. For graph theoretic terminology, we refer Chartrand and Lesniak [5]. Degree of a vertexv is denoted by d(v), the maximum degree of a graph, G is denoted by G. A graph G is connectedif any two vertices of G are connected by a path. A maximal connected sub graph of a graph G is called a component of G. The number of components of G is denoted by ω(g). The complementg of G is the graph with vertex set V in which two vertices are adjacent if and only if they are not adjacent in G. We denote a cycle on p vertices by C p, a pathon p vertices by P p and a complete graph on p vertices by K p. A bipartite graph [or bigraph] is a graph whose vertex set can be divided into two disjoint sets V 1 andv 2, such that every vertex of V 1 is adjacent to every vertex in V 2. A graph G is complete if every pair of its vertices is adjacent. A complete graph on p vertices is denoted by K p. The complete bipartite graph with partitions of order V 1 = mand V 2 = n, is denoted by K m,n. A star, denoted by K 1,p 1 is a tree with one root vertex and p 1 pendant vertices. A cut-vertex (cut edge) of a graph is a vertex (edge) whose removal increases the number of components. A vertex cut (or) separating set of a connected graph G is a set of vertices whose removal results in a disconnected graph. A graph G is regular of degree r if every vertex of G has degree r. Such graphs are called r- regular graphs. Any 3-regular graph is called a cubic graph. Two graphs G 1 and G 2 are isomorphic, written G 1 G 2, if there are bijections θ: V G 1 V G 2 and : E( G 1 307

2 E(G 2 ) such that ψ (e) (φ e G1 = uvifandonlyifψ )) G2 = θ u θ(v) such a pair of mappings is called isomorphism between G 1 and G 2. In our literature survey, we are able to find many authors have introduced various new parameters by imposing conditions on the dominating sets. In that sequence, the concept of connectedness plays an important role in any network. A subset S of V of a nontrivial graph G is called a dominating setof G if every vertex in V S is adjacent to at least one vertex in S. The domination number γ(g) is the minimum cardinality of a dominating set. The concept of triple connected graphs was introduced by Paulraj Joseph et.al [11]. A graph is said to be triple connected if any three vertices lie on a path in G. In [4] the authors introduced triple connected domination number of a graph. A subset S of V of a nontrivial graph G is said to be triple connected dominating set, if S is a dominating set and <S> is triple connected. The minimum cardinality taken over all triple connected dominating sets is called the triple connected domination number of G and is denoted by (G). In [5], the authors Mahadevan et.al introduced the concept of complementary triple connected domination number of a graph. A subset S of V of a nontrivial graph G is said to be complementary triple connected dominating set, if S is a dominating set and the induced sub graph <V-S> is triple connected. The minimum cardinality taken over all complementary triple connected dominating sets is called the complementary triple connected domination number of G and is denoted by c (G). In [8], the authors have introduced Neighborhood triple connected domination number of a graph. A subset S of V of a nontrivial graph G is said to be aneighborhood triple connected dominating set, if S is a dominating set and the induced sub graph<n(s)> is a triple connected. The minimum cardinality taken over all neighborhood triple connecteddominating sets is called the neighborhood triple connected domination number and is denoted by γ n (G). In [9], the authors introduced triple connected.com dominating set. A subset S of V of a nontrivial connected graph G is said to be a triple connected.com dominating set, if S is a triple connected dominating set and the induced sub graph< V-S > is connected. The minimum cardinality taken over all triple connected.com dominating sets is called the triple connected.com domination number and is denoted by, ( ). com G. In [10], A.Ahilaet.al., introducedblast Domination Number of a graph with a real life application. A subset S of V of a non-trivial graph G is said to be a Blast Dominating Set, if S is a connected dominating set and the induced sub graph V S is triple connected. The minimum cardinality taken over all Blast Dominating sets is called the Blast Domination Number and is denoted byγ c G. Notation 1.1: 1. K m P n is the graph got by attaching end vertices P n in any one vertex of K m 2. K m np r is the graph got by attaching n-times P r in any one vertex of K m. 3. K m K n the graph obtained by attaching K n in any one vertex of K m 4. K m C n the graph obtained by attaching C n in any one vertex of K m 2. PRELIMINARIES ABSOLUTE BDN VALUES FOR FEW STANDARD GRAPHS a) Every Complete graph of order,p 4, we get γ c K p = 1, in [10]. b) Theorem: 3.3in [10],For any non-trivial connected graph G, with number of vertices greater than or equal to 4, we get 1 γ c G p 3 and the bounds are sharp. 308

3 3. BLAST DOMINATION NUMBER EQUALS CHROMATIC NUMBER IN 3- REGULAR GRAPHS 3-regular graphsof order 8 Theorem 3.1: Let G be a non-trivial connected 3-regular graph on 8 vertices. Then γ c G = χ G = 3 if and only if G is isomorphic to any one of the graphs in Figure 3.1. v 5 v 5 x z x z v 5 y y z y x G 1 G 2 G 3 Figure 3.1 Proof:Let S = x, y, z be a minimum Blast Domination set of G and V S =,,,, v 5. Since S is a connected dominating set and V S is triple connected, moreover G is 3-regular, the only possible case of S = P 3 and V S = P 5. Therefore, without loss of generality, let y be adjacent to x and z. Let x be adjacent to, ; w be adjacent to andv 5 ; Since S is a dominating set, y must be adjacent to. Now is adjacent to andv 5 ; Also z adjacent to andv 5. Since G is 3-regular, should be adjacent to v 5, so that G G 1 of figure 3.1. If is adjacent to andv 5 ; x adjacent to and ; y adjacent to ; Since S is a dominating set, z adjacent to andv 5. As G is 3-regular, should adjacent v 5, so that G G 2 of figure 3.1. If x, y, z are adjacent to,, respectively, then as S is a dominating set, x (or z) must dominate either (orv 5 ). Now as G is 3-regular, should adjacent v 5, so that G G 3 of figure 3.1. In all the other cases, no graph exists. 3-regular graphs of order 10 Theorem 3.2: Let G be a non-trivial connected 3-regular graph on 10 vertices. Then there exists no graph for which,γ c Proof:By contradiction, let S = x, y, z be a minimum Blast Domination set of G then V S =,,,, v 5, v 6, v 7. As the induced sub graph V s is triple connected, moreover G is 3-regular, the only possible case of S = P 3 and V S = P 7.Clearly, we observe that γ c G 3 for any 3-regular graph of order 10. Hence there exists no 3-regular graph of order 10 for which γ c 3-regular graphs of order 12 Theorem 3.3: Let G be a non-trivial connected 3-regular graph on 12 vertices. Then there exists no graph for which γ c Proof:By contradiction, let S = x, y, z be a minimum Blast Domination set of G and V S =,,,, v 5, v 6, v 7, v 8, v 9. Since the induced sub graph V s is triple connected, moreover G is 3-regular, the only possible case of S = P 3 and V S = P 9.Obviously, we observe that γ c G 3 for any 3-regular graph of order 12. Hence there exists no 3-regular graph of order 12 for which γ c 309

4 4. The Nordhous Gaddum Type results is given below: Theorem: 4.1 Let G be a non-trivial connected graph such that GandG have no isolated vertices and of order p 4. Then (i)γ c G + γ c G 2p 6 and (ii)γ c G γ c G p 3 2. Proof:The bound directly follows from Theorem: 3.3, in [10]. From the following example, we get the Blast Dominating Set of G 4 is S =, v 5, v 6 and that of G =,, v 5. Thus the bound holds the sharpness. v 6 v 5 5. RELATIONSHIP WITH OTHER GRAPH THEORETICAL PARAMETERS: G 4 Theorem: 5.1 For any non-trivial connected graph G, with number of vertices, p 4, γ c G + χ G 2p 3 and equality holds iff G K 4 Proof:Let G be a non-trivial connected graph with number of vertices,p 4. Let γ c G + χ G = 2p 3. Then, γ c G = p 3 andχ G = p, is the only possible case. Since, χ G = p, G K p, but for K p, we have γ c K p = 1, in [10].So that, when= 4, χ G = 4. Therefore,equality holds iffg K 4. Converse is obvious. Theorem: 5.2 For any non-trivial connected graph G, γ c G + χ G = 2p 4 iffg K 5, K 4 P 2 Proof:Let γ c G + χ G = 2p 4. Then the possible cases are (i) γ c G = p 4 andχ G = p (ii) γ c G = p 3 andχ G = p 1 Case (i): Let γ c G = p 4 andχ G = p Since, χ G = p, G K p. But, γ c k p = 1 andsowegetp = 5. Hence, G K 5. Case (ii): Letγ c G = p 3 andχ G = p 1 Since, χ G = p 1, G contains a clique of orderp 1 vertices. Therefore, there exists a vertex,v V, which is adjacent to some vertex in K p 1, say u i. Then v, u i will be the γ c - set. Therefore, γ c G = 2 andencep = 5, sotatk = k 4. If v = 1,If we consider thatv adjacent to u i K 4, then G K 4 P 2 If (v) 2, then γ c G < 2, which is a contradiction. Hence no graph exists. The converse is obvious. Theorem:5.3For any non-trivial connected graph G,γ c G + χ G = 2p 5 iff G K 6, K 5 P 2, K 4 P 3, G 8, G 9, K 4 K 3 or K 4 C 3, K 4 2P 2, 0,0,0 Proof:Letγ c G + χ G = 2p 5. Then the possible cases are 310

5 (i) γ c G = p 5 andχ G = p (ii) γ c G = p 4 andχ G = p 1 (iii) γ c G = p 3 andχ G = p 2 Case (i): Let γ c G = p 5 andχ G = p. Since, χ G = p, G K p. But,γ c k p = 1. Therefore we get p = 6. Hence, G K 6. Case (ii): Letγ c G = p 4 andχ G = p 1. Since, χ G = p 1, G contains a clique K p 1 of orderp 1 vertices. Therefore there exists a vertex,v S, such that vis adjacent to K p 1 and so we get γ c G = 2. Hence, whenp = 6,K = K 5. If d v = 1, then the vertex v S, is adjacent to only one vertex in K 5. We get, γ c G = 2 andχ G = 5. Hence, G K 5 P 2. If d v 2,then γ c G < 2and this is a contradiction. Hence no graph exists. Case (iii): Letγ c G = p 3 andχ G = p 2. Since, χ G = p 2, G contains a clique on p 2 vertices. Then we take S =, such that S = K 2, K 2 Sub case (i): Let S = K 2. Since G is connected, there exists a vertex say, S, adjacent to some u i K p 2. Then,, u i be a γ c - set such that γ c (G) = 3 and hence,p = 6. Hence we get,k = K 4. If d = 2, d = 1,then let ofk 2 be adjacent to any one vertex u 1 ofk 4. Then we get, γ c (G) = 3 and χ G = 4 such thatγ c G + χ G = 2p 5. Therefore, G K 4 P 3. If d = 3 or 4andd = 1, thensuppose we find a vertex K 2, adjacent to two distinct vertices of K 4, we get G G 5.Also, if we raise degree of as 3, by dominating 3 distinct vertices of K 4, we get G G 6. But, if we raise both d( ) > 2 and d( ) > 2, we get γ c G < 3, which leads to contradiction. Hence no graph exists. u 1 u 2 u 1 u 2 u 3 u 4 G 5 u 3 u 4 G 6 If d = d = 2, Suppose, if both & of K 2 adjacent with a single vertex u 1 ofk 4, which satisfies the condition wenp = 6. Hence, we get G K 4 K 3 or K 4 C 3. Suppose, & of K 2 are adjacent to distinct vertices of K 4, we get γ c G < 3, which brings a contradiction. Hence no graph exists. Sub case (ii): Let S = K 2 Suppose, both & of K 2 are adjacent to the same single vertex u 1 ofk 4, we get G K 4 2P 2, 0,0,0. On the other hand, if we raise bothd( ) & d( ) of K 2 is above 1, but γ c G reduces below 3, there exists no graph. Similarly, if both d = d = 1 and both vertices are adjacent to distinct vertices of K 4, then γ c G increases above 3. This also leads to contradiction. Hence no graph exists. The converse is obvious. 311

6 REFERENCES [1] Harary.F Graph Theory, Addison Wesley Reading Mass (1972) [2] T.W. Haynes,S. T.Hedetniemi and P.J.Slater, Fundamentals of Domination ingraphs, Marcel Dekker, Inc.,New York,1998. [3] E.Sampathkumar and H.B.Walikar, The connected domination number of a graph, J.Math.Phy.Sci., 13(6) (1979), [4] G.Mahadevan, A.Selvam, J.Paulraj Joseph and T.Subramanian, Triple connected domination number of a graph, International Journal of Mathematical Combinatorics, Vol.3 (2012), [5] G.Mahadevan, A.Selvam, J.Paulraj Joseph, B.Ayisha and T.Subramanian, Complementary triple connected domination number of a graph, Advances and Applications in Discrete Mathematics, Vol. 12 (I) (2013), [6] G.Mahadevan, A.Selvam, A.Mydeen Bibi and T.Subramanian, Complementary perfect triple connected domination number of a graph, International Journal of Engineering Research and Application, Vol.2, Issue 5 (2012), [7] G.Mahadevan, A.Selvam, A.Nagarajan, A.Rajeswari and T.Subramanian, Paired Triple connected domination number of a graph, International Journal of Computational Engineering Research, Vol. 2, Issue 5 (2012), [8] G.Mahadevan, A.Ahila, N.Ramesh, C.Sivagnanam, Selvam Avadayappan and T.Subramanian, Neighborhood triple connected domination number of a graph, International Journal of Computational Engineering Research, Vol, 04, Issue, 3, [9] G.Mahadevan, M.Lavanya, Selvam Avadayappan and T.Subramanian, Triple connected.com domination number of a graph, International Journal of Mathematical Archive 4 (10), 2013, [10] G.Mahadevan, A.Ahila and Selvam Avadayappan, Blast Domination Number of a Graph Preprint. [11] J.Paulraj Joseph, M.K.Angel Jebitha, P.Chithra Devi and G.Sudhana, Triple connected graphs, Indian Journal of Mathematics and Mathematical Sciences, Vol.8, No.I(2012),

Triple Connected Domination Number of a Graph

International J.Math. Combin. Vol.3(2012), 93-104 Triple Connected Domination Number of a Graph G.Mahadevan, Selvam Avadayappan, J.Paulraj Joseph and T.Subramanian Department of Mathematics Anna University: