Triple Connected Complementary Tree Domination Number of a Graph

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1 International Mathematical Form, Vol. 8, 2013, no. 14, HIKARI Ltd, Triple Connected Complementar Tree Domination Nmber of a Graph G. Mahadevan 1, Selvam Avadaappan 2, N. Ramesh 3 and T. Sbramanian 4 1 Dept. of Mathematics, Gandhigram Rral Institte Deemed Universit, Gandhigram , Tamilnad, India Crrent address: Department of Mathematics, Anna Universit: Tirnelveli Region, Tirnelveli , Tamilnad, India gmaha2003@ahoo.co.in 2 Dept.of Mathematics, VHNSN College Virdhnagar , Tamilnad, India selvam_avadaappan@ahoo.co.in 3 Udaa School of Engineering, Udaa Nagar Vellamodi, Kanakmari , Tamilnad, India nrameshnair@ahoo.com 4 Dept. of Mathematics, Anna Universit: Tirnelveli Region, Tirnelveli , Tamilnad, India tsbramanian001@gmail.com Copright 2013 G. Mahadevan et al. This is an open access article distribted nder the Creative Commons Attribtion License, which permits nrestricted se, distribtion, and reprodction in an medim, provided the original work is properl cited. Abstract The concept of triple connected graphs with real life application was introdced in [14] b considering the existence of a path containing an three vertices of a graph G. In [4], G. Mahadevan et. al., introdced triple connected domination nmber of a graph. A sbset S of V of a nontrivial connected graph G is said to be triple connected dominating set, if S is a dominating set and the indced sb graph <S> is triple connected. The minimm cardinalit taken over all triple connected dominating sets is called the triple connected domination nmber of G and is denoted b tc(g). A sbset S of V of a nontrivial graph connected graph G is said to be a complementar tree dominating set, if S is a dominating set and the indced sb graph <V - S> is a tree. The minimm cardinalit taken over all

2 660 G. Mahadevan et al complementar tree dominating sets is called the complementar tree domination nmber of G and is denoted b ctd(g). In this paper we introdce a new domination parameter, called triple connected complementar tree domination nmber of a graph. A sbset S of V of a nontrivial connected graph G is said to be triple connected complementar tree dominating set, if S is a triple connected dominating set and the indced sb graph <V - S> is a tree. The minimm cardinalit taken over all triple connected complementar tree dominating sets is called the triple connected complementar tree domination nmber of G and is denoted b tct(g). We determine this nmber for some standard graphs and obtain bonds for general graphs. Its relationship with other graph theoretical parameters are also investigated. Mathematics Sbject Classification: 05C69 Kewords: Domination Nmber, Triple connected graph, Triple connected domination nmber, Triple connected complementar tree domination nmber. 1. Introdction B a graph we mean a finite, simple, connected and ndirected graph G(V, E), where V denotes its vertex set and E its edge set. Unless otherwise stated, the graph G has p vertices and q edges. Degree of a vertex v is denoted b d(v), the maximm degree of a graph G is denoted b Δ(G). We denote a ccle on p vertices b C p, a path on p vertices b P p, and a complete graph on p vertices b K p. A graph G is connected if an two vertices of G are connected b a path. A maximal connected sbgraph of a graph G is called a component of G. The nmber of components of G is denoted b (G). The complement of G is the graph with vertex set V in which two vertices are adjacent if and onl if the are not adjacent in G. A tree is a connected acclic graph. A bipartite graph (or bigraph) is a graph whose vertex set can be divided into two disjoint sets V 1 and V 2 sch that ever edge has one end in V 1 and another end in V 2. A complete bipartite graph is a bipartite graph where ever vertex of V 1 is adjacent to ever vertex in V 2. The complete bipartite graph with partitions of order V 1 =m and V 2 =n, is denoted b K m,n. A star, denoted b K 1,p-1 is a tree with one root vertex and p 1 pendant vertices. A bistar, denoted b B(m, n) is the graph obtained b

3 Triple connected complementar tree domination nmber 661 joining the root vertices of the stars K 1,m and K 1,n. The friendship graph, denoted b F n can be constrcted b identifing n copies of the ccle C 3 at a common vertex. A wheel graph, denoted b W p is a graph with p vertices, formed b connecting a single vertex to all vertices of C p-1. A helm graph, denoted b H n is a graph obtained from the wheel W n b attaching a pendant vertex to each vertex in the oter ccle of W n. Corona of two graphs G 1 and G 2, denoted b G 1 G 2 is the graph obtained b taking one cop of G 1 and V 1 copies of G 2 ( V 1 is the nmber of vertices in G 1 ) in which i th vertex of G 1 is joined to ever vertex in the i th cop of G 2. If S is a sbset of V, then <S> denotes the vertex indced sbgraph of G indced b S. The open neighborhood of a set S of vertices of a graph G, denoted b N(S) is the set of all vertices adjacent to some vertex in S and N(S) S is called the closed neighborhood of S, denoted b N[S]. The diameter of a connected graph is the maximm distance between two vertices in G and is denoted b diam(g). A ct vertex (ct edge) of a graph G is a vertex (edge) whose removal increases the nmber of components. A vertex ct, or separating set of a connected graph G is a set of vertices whose removal reslts in a disconnected graph. The connectivit or vertex connectivit of a graph G, denoted b κ(g) (where G is not complete) is the sie of a smallest vertex ct. A connected sbgraph H of a connected graph G is called a H -ct if (G H) 2. The chromatic nmber of a graph G, denoted b χ(g) is the smallest nmber of colors needed to color all the vertices of a graph G in which adjacent vertices receive different colors. For an real nmber, denotes the largest integer less than or eqal to. A Nordhas -Gaddm-tpe reslt is a (tight) lower or pper bond on the sm or prodct of a parameter of a graph and its complement. Terms not defined here are sed in the sense of [11]. A sbset S of V is called a dominating set of G if ever vertex in V S is adjacent to at least one vertex in S. The domination nmber (G) of G is the minimm cardinalit taken over all dominating sets in G. A dominating set S of a connected graph G is said to be a connected dominating set of G if the indced

4 662 G. Mahadevan et al sb graph <S> is connected. The minimm cardinalit taken over all connected dominating sets is the connected domination nmber and is denoted b c. Man athors have introdced different tpes of domination parameters b imposing conditions on the dominating set [3, 16]. Recentl, the concept of triple connected graphs has been introdced b Palraj Joseph J. et. al.,[14] b considering the existence of a path containing an three vertices of G. The have stdied the properties of triple connected graphs and established man reslts on them. A graph G is said to be triple connected if an three vertices lie on a path in G. All paths, ccles, complete graphs and wheels are some standard examples of triple connected graphs. In [4] Mahadevan G. et. al., introdced triple connected domination nmber of a graph and fond man reslts on them. A sbset S of V of a nontrivial connected graph G is said to be triple connected dominating set, if S is a dominating set and the indced sb graph <S> is triple connected. The minimm cardinalit taken over all triple connected dominating sets is called the triple connected domination nmber of G and is denoted b tc(g). In [5, 6, 7, 8, 9, 10] Mahadevan G. et. al., introdced complementar triple connected domination nmber, paried triple connected domination nmber, complementar perfect triple connected domination nmber, triple connected two domination nmber, restrained triple connected domination nmber, dom strong triple connected domination nmber of a graph. A sbset S of V of a nontrivial graph connected graph G is said to be a complementar tree dominating set, if S is a dominating set and the indced sb graph <V - S> is a tree. The minimm cardinalit taken over all complementar tree dominating sets is called the complementar tree domination nmber of G and is denoted b ctd(g). In this paper, we se this idea to develop the concept of triple connected complementar tree dominating set and triple connected complementar tree domination nmber of a graph. Theorem 1.1 [14] A connected graph G is not triple connected if and onl if there exists a H -ct with (G H) 3 sch that ( ) ( ) = 1 for at least three components C 1, C 2, and C 3 of G H.

5 Triple connected complementar tree domination nmber 663 Notation 1.2 Let G be a connected graph with m vertices v1, v2,., vm. The graph obtained from G b attaching n1 times a pendant vertex of times a pendant vertex of n2,., nm, n3 on the vertex v1, n2 on the vertex v2 and so on, is denoted b G(n1, ) where ni, li 0 and 1 i m. Example 1.3 Let v1, v2, v3, v4, be the vertices of C4. The graph C4(P2, 2P2, 3P2, P3) is obtained from C4 b attaching 1 time a pendant vertex of P2 on v1, 2 times a pendant vertex of P2 on v2, 3 times a pendant vertex of P2 on v3 and 1 time a pendant vertex of P3 on v4 and is shown in Figre 1.1. v1 v2 v3 v4 Figre 1.1 : C4(P2, 2P2, 3P2, P3) 2. Triple connected complementar tree domination nmber Definition 2.1 A sbset S of V of a nontrivial connected graph G is said to be a triple connected complementar tree dominating set, if S is a triple connected dominating set and the indced sbgraph <V - S> is a tree. The minimm cardinalit taken over all triple connected complementar tree dominating sets is called the triple connected complementar tree domination nmber of G and is denoted b a tct tct(g). An triple connected dominating set with tct vertices is called -set of G. Example 2.2 For the graph G1 in Figre 2.1, S = {v1, v2, v3} forms a Hence tct(g1) = 3. v1 v2 v3 G 1: v7 v6 v4 v5 Figre 2.1 : Graph with tct = 3. tct -set of G1.

6 664 G. Mahadevan et al Observation 2.3 Triple connected complementar tree dominating set does not exists for all graphs and if exists, then tct(g) 3. Throghot this paper we consider onl connected graphs for which triple connected complementar tree dominating set exists. Observation 2.4 The complement of the triple connected complementar tree dominating set need not be a triple connected complementar tree dominating set. Observation 2.5 Ever triple connected complementar tree dominating set is a dominating set bt not conversel. Observation 2.6 For an connected graph G, c(g) tc(g) tct(g) and for the ccle C 7 the bonds are sharp. Theorem 2.7 If the indced sbgraph of each connected dominating set of G has more than two pendant vertices, then G does not contain a triple connected complementar tree dominating set. Proof The proof follows from Theorem 1.1. Exact vale for some standard graphs: 1) For an ccle of order p 5, tct(c p ) = p 2. 2) For an complete bipartite graph of order p 5, tct(k m,n ) = p - 2. (where m, n 2 and m + n = p ). 3) For an complete graph of order p 5, tct(k p ) = p ) For an wheel of order p 5, tct(w p ) = p - 2. Exact vale for some special graphs: 1) The Moser spindle (also called the Mosers' spindle or Moser graph) is an ndirected graph, named after mathematicians Leo Moser and his brother William, with seven vertices and eleven edges given in Figre 2.2.

7 Triple connected complementar tree domination nmber 665 Figre 2.2 For the Moser spindle graph G, tct(g) = 3. 2) The Wagner graph is a 3-reglar graph with 8 vertices and 12 edges given in Figre 2.3. Figre 2.3 For the Wagner graph G, tct(g) = 4. 3) The Bidiakis cbe is a 3-reglar graph with 12 vertices and 18 edges given in Figre 2.4. Figre 2.4 For the Bidiakis cbe graph G, tct(g) = 8. 4) The Frcht graph is a 3-reglar graph with 12 vertices, 18 edges, and no nontrivial smmetries given in Figre 2.5.

8 666 G. Mahadevan et al Figre 2.5 For the Frcht graph G, tct(g) = 8. Theorem 2.8 For an connected graph G with p 5, we have 3 tct(g) p 2 and the bonds are sharp. Proof The lower and pper bonds follows from Definition 2.1. For C5, the lower bond is attained and for K6 the pper bond is attained. Theorem 2.9 For a connected graph G with 5 vertices, tct(g) = p 2 if and onl if G is isomorphic to C5, W5, K5, K2,3, F2, K5 {e}, K4(P2), C4(P2), C3(P3), C3(2P2) or an one of the graphs shown in Figre 2.6. x H1: v H2: v H3: H4: x x x v H5: v x H6: v x H7: v Figre 2.6 : Graphs with v x tct = p 2.

9 Triple connected complementar tree domination nmber 667 Proof Sppose G is isomorphic to C 5, W 5, K 5, K 2,3, F 2, K 5 {e}, K 4 (P 2 ), C 4 (P 2 ), C 3 (P 3 ), C 3 (2P 2 ) or an one of the graphs H 1 to H 7 given in Figre 2.2., then clearl tct(g) = p 2. Conversel, let G be a connected graph with 5 vertices and tct(g) = 3. Let S = {x,, } be a tct -set, then clearl <S> = P 3 or C 3. Let V S = V(G) V(S) = {, v}, then <V S> = K 2. Case (i) <S> = P 3 = x. Since G is connected and S is a tct -set, there exists a vertex sa x (or ) in P 3 which is adjacent to and v in K 2. Then S = {x,, } forms a tct -set of G so that tct(g) = p 2. If d(x) = d() = 2, d() = 2, then G C 5. Since G is connected and S is a tct -set, there exists a vertex sa in P 3 is adjacent to and v in K 2. Then S = {x,, } forms a tct -set of G so that tct(g) = p 2. If d(x) = d() = 1, d() = 4, then G C 3 (2P 2 ). Now b increasing the degrees of the vertices, b the above argments, we have G W 5, K 5, K 2,3, K 5 {e}, K 4 (P 2 ), C 4 (P 2 ), C 3 (P 3 ), and H 1 to H 7 in Figre 2.2. In all the other cases, no new graph exists. Case (ii) <S> = C 3 = xx. Since G is connected, there exists a vertex sa x (or, ) in C 3 is adjacent to (or v) in K 2. Then S = {x,, v} forms a tct -set of G so that tct(g) = p 2. If d(x) = 3, d() = d() = 2, then G C 3 (P 3 ). If d(x) = 4, d() = d() = 2, then G F 2. In all the other cases, no new graph exists. Theorem 2.10 Ever triple connected complementar tree dominating set mst contains all the pendant vertices. Proof Let v be a vertex of G sch that d(v) = 1 and let S be a tctd set of G. If v is not in S, then a vertex adjacent to v mst be in S and hence <V S> is disconnected, which is a contradiction. Corollar 2.11 If G is a graph with m pendant vertices, then tct(g) m.

10 668 G. Mahadevan et al Proof The proof is directl follows from Theorem The Nordhas Gaddm tpe reslt is given below: Theorem 2.12 Let G be a graph sch that G and have no isolates of order p 5. Then (i) tct(g) + tct( ) 2(p 2) (ii) tct(g). tct( ) 2(p 2) 2 and the bond is sharp. Proof The bond directl follows from Theorem 2.8. For the ccle C 7, tc(g) + tc( ) = 2(p 2) and for K 5, tct(g). tct( ) 2(p 2) 2. 3 Relation with Other Graph Theoretical Parameters Theorem 3.1 For an connected graph G with p 5 tct(g) + κ(g) 2p 3 and the bond is sharp if and onl if G K p. vertices, Proof Let G be a connected graph with p 5 vertices. We know that κ(g) p 1 and b Theorem 2.8, tct(g) p 2. Hence tct(g) + κ(g) 2p 3. Sppose G is isomorphic to K p. Then clearl tct(g) + κ(g) = 2p 3. Conversel, Let tct(g) + κ(g) = 2p 3. This is possible onl if tct(g) = p 2 and κ(g) = p 1. Bt κ(g) = p 1, and so G K p. Theorem 3.2 For an connected graph G with p 5 vertices, tct(g) + (G) 2p 2 and the bond is sharp if and onl if G K p. Proof Let G be a connected graph with p 5 vertices. We know that (G) p and b Theorem 2.8, tct(g) p 2. Hence tct(g) + (G) 2p 2. Sppose G is isomorphic to K p. Then clearl tct(g) + (G) = 2p 2. Conversel, let tct(g) + (G) = 2p 2. This is possible onl if tct(g) = p 2 and (G) = p. Since (G) = p, G is isomorphic to K p.

11 Triple connected complementar tree domination nmber 669 Theorem 3.3 For an connected graph G with p 5 vertices, tct(g) + (G) 2p 3 and the bond is sharp. Proof Let G be a connected graph with p 5 vertices. We know that (G) p 1 and b Theorem 2.8, tct(g) p 2. Hence tct(g) + (G) 2p 3. For K 8, the bond is sharp. References [1] E. A. Nordhas and, J. W. Gaddm, On complementar graphs, Amer. Math. Monthl, 63 (1956), [2] E. J. Cokane and, S. T. Hedetniemi, Total domination in graphs, Networks,Vol.10 (1980), [3] E. Sampathkmar and, H. B. Walikar, The connected domination nmber of a graph, J.Math. Phs. Sci., 13 (6) (1979), [4] G. Mahadevan, A. Selvam, J. Palraj Joseph and, T. Sbramanian, Triple connected domination nmber of a graph, International Jornal of Mathematical Combinatorics, Vol.3 (2012), [5] G. Mahadevan, A. Selvam, J. Palraj Joseph, B. Aisha and, T. Sbramanian, Complementar triple connected domination nmber of a graph, Accepted for pblication in Advances and Applications in Discrete Mathematics, (2012). [6] G. Mahadevan, A. Selvam, A. Mdeen bibi and, T. Sbramanian, Complementar perfect triple connected domination nmber of a graph, International Jornal of Engineering Research and Application, Vol.2, Isse 5 (2012), [7] G. Mahadevan, A. Selvam, A. Nagarajan, A. Rajeswari and, T. Sbramanian, Paired Triple connected domination nmber of a graph, International Jornal of Comptational Engineering Research, Vol. 2, Isse 5 (2012), [8] G. Mahadevan, A. Selvam, B. Aisha, and, T. Sbramanian, Triple connected two domination nmber of a graph, International Jornal of Comptational Engineering Research Vol. 2, Isse 6 (2012), [9] G. Mahadevan, A. Selvam, V. G. Bhagavathi Ammal and, T. Sbramanian, Restrained triple connected domination nmber of a graph, International Jornal of Engineering Research and Application, Vol. 2, Isse 6 (2012),

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