G. Mahadevan 1 Selvam Avadayappan 2 V. G. Bhagavathi Ammal 3 T. Subramanian 4

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1 G. Mahadean, Selam Aadaappan, V. G. Bhagaathi Ammal, T. Sbramanian / International Jornal of Engineering Research and Applications (IJERA) ISSN: Vol. 2, Isse 6, Noember- December 2012, pp Restrained Triple Connected Domination Nmber of a Graph G. Mahadean 1 Selam Aadaappan 2 V. G. Bhagaathi Ammal 3 T. Sbramanian 4 1,4 Dept. of Mathematics, Anna Uniersit: Tirneleli Region, Tirneleli. 2 Dept.of Mathematics,VHNSN College, Virdhnagar. 3 PG Dept.of Mathematics, Sree Aappa College for Women, Chnkankadai, Nagercoil. Abstract The concept of triple connected graphs with real life application was introdced in [9] b considering the eistence of a path containing an three ertices of a graph G. In[3], G. Mahadean et. al., was introdced the concept of triple connected domination nmber of a graph. In this paper, we introdce a new domination parameter, called restrained triple connected domination nmber of a graph. A sbset S of V of a nontriial graph G is called a dominating set of G if eer erte in V S is adjacent to at least one erte in S. The domination nmber (G) of G is the minimm cardinalit taken oer all dominating sets in G. A sbset S of V of a nontriial graph G is called a restrained dominating set of G if eer erte in V S is adjacent to at least one erte in S as well as another erte in V - S. The restrained domination nmber r (G) of G is the minimm cardinalit taken oer all restrained dominating sets in G. A sbset S of V of a nontriial 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 oer all triple connected dominating sets is called the triple connected domination nmber and is denoted b γ tc. A sbset S of V of a nontriial graph G is said to be restrained triple connected dominating set, if S is a restrained dominating set and the indced sb graph <S> is triple connected. The minimm cardinalit taken oer all restrained triple connected dominating sets is called the restrained triple connected domination nmber and is denoted b γ rtc. We determine this nmber for some standard graphs and obtain bonds for general graph. Its relationship with other graph theoretical parameters are also inestigated. Ke words: Domination Nmber, Triple connected graph, Triple connected domination nmber, Restrained Triple connected domination nmber. AMS (2010): 05C69 1. Introdction B a graph we mean a finite, simple, connected and ndirected graph G(V, E), where V denotes its erte set and E its edge set. Unless otherwise stated, the graph G has p ertices and q edges. Degree of a erte is denoted b d(), the maimm degree of a graph G is denoted b Δ(G). We denote a ccle on p ertices b C p, a path on p ertices b P p, and a complete graph on p ertices b K p. A graph G is connected if an two ertices of G are connected b a path. A maimal connected sbgraph of a graph G is called a component of G. The nmber of components of G is denoted b ω(g). The complement G of G is the graph with erte set V in which two ertices 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 erte set can be diided into two disjoint sets V 1 and V 2 sch that eer edge has one end in V 1 and another end in V 2. A complete bipartite graph is a bipartite graph where eer erte of V 1 is adjacent to eer erte 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 erte and p 1 pendant ertices. A bistar, denoted b B(m, n) is the graph obtained b joining the root ertices 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 erte. A wheel graph, denoted b W p is a graph with p ertices, formed b connecting a single erte to all ertices of C p-1. If S is a sbset of V, then <S> denotes the erte indced sbgraph of G indced b S. The open neighborhood of a set S of ertices of a graph G, denoted b N(S) is the set of all ertices adjacent to some erte in S and N(S) S is called the closed neighborhood of S, denoted b N[S]. A ct erte (ct edge) of a graph G is a erte (edge) whose remoal increases the nmber of components. A erte ct, or separating set of a connected graph G is a set of ertices whose remoal reslts in a disconnected. The connectiit or erte connectiit of a graph G, denoted b κ(g) (where G is not complete) is the sie of a smallest erte ct. The chromatic nmber of a graph G, denoted b χ(g) is the smallest nmber of colors needed to color all the ertices of a graph G in which adjacent ertices receie 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 225 P a g e

2 G. Mahadean, Selam Aadaappan, V. G. Bhagaathi Ammal, T. Sbramanian / International Jornal of Engineering Research and Applications (IJERA) ISSN: Vol. 2, Isse 6, Noember- December 2012, pp 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 [2]. A sbset S of V is called a dominating set of G if eer erte in V S is adjacent to at least one erte in S. The domination nmber (G) of G is the minimm cardinalit taken oer 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 sb graph <S> is connected. The minimm cardinalit taken oer all connected dominating sets is the connected domination nmber and is denoted b c. A dominating set is said to be restrained dominating set if eer erte in V S is adjacent to atleast one erte in S as well as another erte in V - S. The minimm cardinalit taken oer all restrained dominating sets is called the restrained domination nmber and is denoted b γ r. Man athors hae introdced different tpes of domination parameters b imposing conditions on the dominating set [11, 12]. Recentl, the concept of triple connected graphs has been introdced b J. Palraj Joseph et. al.,[9] b considering the eistence of a path containing an three ertices of G. The hae 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 ertices lie on a path in G. All paths, ccles, complete graphs and wheels are some standard eamples of triple connected graphs. In[3], G. Mahadean et. al., was introdced the concept of triple connected domination nmber of a graph. A sbset S of V of a nontriial graph G is said to be a triple connected dominating set, if S is a dominating set and the indced sbgraph <S> is triple connected. The minimm cardinalit taken oer all triple connected dominating sets is called the triple connected domination nmber of G and is denoted b γ tc (G). An triple connected dominating set with γ tc ertices is called a γ tc -set of G. In[4, 5, 6], G. Mahadean et. al., was introdced complementar triple connected domination nmber, complementar perfect triple connected domination nmber and paired triple connected domination nmber of a graph and inestigated new reslts on them. In this paper, we se this idea to deelop the concept of restrained triple connected dominating set and restrained triple connected domination nmber of a graph. Theorem 1.1 [9] A tree T is triple connected if and onl if T P p ; p 3. Notation 1.2 Let G be a connected graph with m ertices 1, 2,., m. The graph obtained from G b attaching n 1 times a pendant erte of P l1 on the erte 1, n 2 times a pendant erte of P l2 on the erte 2 and so on, is denoted b G(n 1 P l1, n 2 P l2, n 3 P l3,., n m P lm ) where n i, l i 0 and 1 i m. Eample 1.3 Let 1, 2, 3, 4, be the ertices of K 4. The graph K 4 (2P 2, 2P 2, 2P 3, P 3 ) is obtained from K 4 b attaching 2 times a pendant erte of P 2 on 1, 2 times a pendant erte of P 2 on 2, 2 times a pendant erte of P 3 on 3 and 1 time a pendant erte of P 3 on 4 and is shown in Figre Figre 1.1 : K 4 (2P 2, 2P 2, 2P 3, P 3 ) 2. Restrained Triple connected domination nmber Definition 2.1 A sbset S of V of a nontriial graph G is said to be a restrained triple connected dominating set, if S is a restrained dominating set and the indced sbgraph <S> is triple connected. The minimm cardinalit taken oer all restrained triple connected dominating sets is called the restrained triple connected domination nmber of G and is denoted b γ rtc (G). An triple connected two dominating set with γ rtc ertices is called a γ rtc - set of G. Eample 2.2 For the graphs G 1, G 2, G 3 and G 4, in Figre 2.1, the hea dotted ertices forms the restrained triple connected dominating sets. Figre 2.1 : Graph with γ rtc = 3. Obseration 2.3 Restrained Triple connected dominating set (rtcd set) does not eists for all graphs and if eists, then γ rtc (G) P a g e

3 G. Mahadean, Selam Aadaappan, V. G. Bhagaathi Ammal, T. Sbramanian / International Jornal of Engineering Research and Applications (IJERA) ISSN: Vol. 2, Isse 6, Noember- December 2012, pp Eample 2.4 For the graph G 5, G 6 in Figre 2.2, we cannot find an restrained triple connected dominating set. Eample 2.11 Consider the graph G 8 and its spanning sbgraph H 8 and G 9 and its spanning sbgraph H 9 shown in Figre G 8 : H 8 : Figre 2.2 : Graphs with no rtcd set Throghot this paper we consider onl connected graphs for which triple connected two dominating set eists. Obseration 2.5 The complement of the restrained triple connected dominating set need not be a restrained triple connected dominating set. Eample 2.6 For the graph G 7 in Figre 2.3, S = { 1, 4, 2 } forms a restrained triple connected dominating set of G 3. Bt the complement V S = { 3, 5 } is not a restrained triple connected dominating set. G 7 : Figre 2.3 : Graph in which V S is not a rtcd set Obseration 2.7 Eer restrained triple connected dominating set is a dominating set bt not conersel. Obseration 2.8 Eer restrained triple connected dominating set is a connected dominating set bt not conersel. Eact ale for some standard graphs: 1) For an ccle of order p 5, γ rtc (C p ) = p 2. 2) For an complete graph of order p 5, γ rtc (K p ) = 3. 3) For an complete bipartite graph of order p 5, γ rtc (K m,n ) = 3. (where m, n 2 and m + n = p ). Obseration 2.9 If a spanning sb graph H of a graph G has a restrained triple connected dominating set, then G also has a restrained triple connected dominating set. Obseration 2.10 Let G be a connected graph and H be a spanning sb graph of G. If H has a restrained triple connected dominating set, then γ rtc (G) γ rtc (H) and the bond is 2 5 G 9 : H 9 : Figre 2.4 For the graph G 8, S = { 1, 4, 2 } is a restrained triple connected dominating set and so γ rtc (G 4 ) = 3. For the spanning sbgraph H 8 of G 8, S = { 1, 4, 2, 5 } is a restrained triple connected dominating set so that γ rtc (H 8 ) = 4. Hence γ rtc (G 8 ) < γ rtc (H 8 ). For the graph G 9, S = { 1, 2, 3 } is a restrained triple connected dominating set and so γ rtc (G 9 ) = 3. For the spanning sbgraph H 9 of G 9, S = { 1, 2, 3 } is a restrained triple connected dominating set so that γ rtc (H 9 ) = 3.Hence γ rtc (G 9 ) = γ rtc (H 9 ). Theorem 2.12 For an connected graph G with p 5, we hae 3 γ rtc (G) p - 2 and the bonds are Proof The lower and bonds follows from Definition 2.1. For K 6, the lower bond is attained and for C 9 the pper bond is attained. Theorem 2.13 For an connected graph G with 5 ertices, γ rtc (G) = p - 2 if and onl if G K 5, C 5, F 2, K 5 e, K 4 (P 2 ), C 4 (P 2 ), C 3 (P 3 ), C 3 (2P 2 ) and an one of the following graphs gien in Figre P a g e

4 G. Mahadean, Selam Aadaappan, V. G. Bhagaathi Ammal, T. Sbramanian / International Jornal of Engineering Research and Applications (IJERA) ISSN: Vol. 2, Isse 6, Noember- December 2012, pp G 1 : G 2 : G 3 : G 4 : G 5 : G 6 : G 7 : G 8 : G 9 : Figre 2.5 Proof Sppose G is isomorphic to K 5, C 5, F 2, K 5 e, K 4 (P 2 ), C 4 (P 2 ), C 3 (P 3 ), C 3 (2P 2 ) and an one of the gien graphs in Figre 2.5., then clearl γ rtc (G) = p - 2. Conersel, Let G be a connected graph with 5 ertices, and γ rtc (G) = 3. Let S = {,, } be the γ rtc (G) set of G. Take V S = {, } and hence <V S> = K 2 =. Case (i) <S> = P 3 =. Since G is connected, (or eqialentl ) is adjacent to (or eqialentl ) (or) is adjacent to (or eqialentl ). If is adjacent to. Since S is a restrained triple connected dominating set, is adjacent to (or) (or). If is adjacent to, then G C 5. If is adjacent to, then G C 4 (P 2 ). Now b increasing the degrees of the ertices of K 2 =, we hae G G 1 to G 5, K 5 e, C 3 (P 3 ). Now let be adjacent to. Since S is a restrained triple connected dominating set, is adjacent to (or) (or). If is adjacent to, then G C 3 (2P 2 ). If is adjacent to and, is adjacent to, then G K 4 (P 2 ). Now b increasing the degrees of the ertices, we hae G G 6 to G 8, C 3 (2P 2 ). Case (ii) <S> = C 3 =. Since G is connected, there eists a erte in C 3 sa is adjacent to (or). Let be adjacent to. Since S is a restrained triple connected dominating set, is adjacent to, then G F 2. Now b increasing the degrees of the ertices, we hae G G 9, K 5. In all the other cases, no new graph eists. The Nordhas Gaddm tpe reslt is gien below: Theorem 2.16 Let G be a graph sch that G and G hae no isolates of order p 5. Then (i) γ rtc (G) + γ rtc (G) 2p -4 (ii) γ rtc (G). γ rtc (G) (p 2) 2 and the bond is Proof The bond directl follows from Theorem For ccle C 5, both the bonds are attained. 3 Relation with Other Graph Theoretical Parameters Theorem 3.1 For an connected graph G with p 5 ertices, γ rtc (G) + κ(g) 2p 3 and the bond is sharp if and onl if G K 5. ertices. We know that κ(g) p 1 and b Theorem 2.12, γ rtc (G) p 2. Hence γ rtc (G) + κ(g) 2p 3. Sppose G is isomorphic to K 5. Then clearl γ rtc (G) + κ(g) = 2p 3. Conersel, Let γ rtc (G) + κ(g) = 2p 3. This is possible onl if γ rtc (G) = p 2 and κ(g) = p 1. Bt κ(g) = p 1, and so G K p for which γ rtc (G) = 3 = p 2. Hence G K 5. Theorem 3.2 For an connected graph G with p 5 ertices, γ rtc (G) + χ(g) 2p 2 and the bond is sharp if and onl if G K 5. ertices. We know that χ(g) p and b Theorem 2.12, γ rtc (G) p 2. Hence γ rtc (G) + χ(g) 2p 2. Sppose G is isomorphic to K 5. Then clearl γ rtc (G) + χ(g) = 2p 2. Conersel, let γ rtc (G) + χ(g) = 228 P a g e

5 G. Mahadean, Selam Aadaappan, V. G. Bhagaathi Ammal, T. Sbramanian / International Jornal of Engineering Research and Applications (IJERA) ISSN: Vol. 2, Isse 6, Noember- December 2012, pp p 2. This is possible onl if γ rtc (G) = p 2 and χ(g) = p. Since χ(g) = p, G is isomorphic to K p for which γ rtc (G) = 3 = p 2. Hence G K 5. Theorem 3.3 For an connected graph G with p 5 ertices, γ rtc (G) + (G) 2p 3 and the bond is ertices. We know that (G) p 1 and b Theorem 2.12, γ rtc (G) p. Hence γ rtc (G) + (G) 2p 3. For K 5, the bond is REFERENCES [1] Cokane E. J. and Hedetniemi S. T. (1980): Total domination in graphs, Networks, Vol.10: [2] John Adrian Bond, Mrt U.S.R. (2009): Graph Theor, Springer, [3] Mahadean G., Selam A., Palraj Joseph J., and Sbramanian T. (2012): Triple connected domination nmber of a graph, IJMC, Vol.3. [4] Mahadean G., Selam A., Palraj Joseph J., Aisha B., and Sbramanian T. (2012): Complementar triple connected domination nmber of a graph, Accepted for pblication in Adances and Applications in Discrete Mathematics, ISSN [5] Mahadean G, Selam Aadaappan, Mdeen bibi A., Sbramanian T. (2012): Complementar perfect triple connected domination nmber of a graph, International Jornal of Engineering Research and Application, ISSN , Vol.2, Isse 5,Sep Oct, pp [6] Mahadean. G, Selam Aadaappan, Nagarajan. A, Rajeswari. A, Sbramanian. T. (2012): Paired Triple connected domination nmber of a graph, International Jornal of Comptational Engineering Research, Vol. 2, Isse 5, Sep. 2012, pp [7] Nordhas E. A. and Gaddm J. W. (1956): On complementar graphs, Amer. Math. Monthl, 63: [8] Palraj Joseph J. and Armgam. S. (1997): Domination and coloring in graphs, International Jornal of Management Sstems, 8 (1): [9] Palraj Joseph J., Angel Jebitha M.K., Chithra Dei P. and Sdhana G. (2012): Triple connected graphs, Indian Jornal of Mathematics and Mathematical Sciences, ISSN Vol. 8, No. 1: [10] Palraj Joseph J. and Mahadean G. (2006): On complementar perfect domination nmber of a graph, Acta Ciencia Indica, Vol. XXXI M, No. 2.: [11] Sampathkmar, E.; Walikar, HB (1979): The connected domination nmber of a graph, J. Math. Phs. Sci 13 (6): [12] Teresa W. Hanes, Stephen T. Hedetniemi and Peter J. Slater (1998): Domination in graphs, Adanced Topics, Marcel Dekker, New York. [13] Teresa W. Hanes, Stephen T. Hedetniemi and Peter J. Slater (1998): Fndamentals of domination in graphs, Marcel Dekker, New York. 229 P a g e