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G. Mahadevan, Selvam Avadayappan, V. G. Bhagavathi Ammal, T. Subramanian / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.225-229 Restrained Triple Connected Domination Number of a Graph G. Mahadevan1 Selvam Avadayappan 2 V. G. Bhagavathi Ammal3 T. Subramanian4 1,4 Dept. of Mathematics, Anna University: Tirunelveli Region, Tirunelveli. 2 Dept.of Mathematics,VHNSN College, Virudhunagar. 3 PG Dept.of Mathematics, Sree Ayyappa College for Women, Chunkankadai, Nagercoil.Abstract The concept of triple connected graphs denotes its vertex set and E its edge set. Unlesswith real life application was introduced in [9] by otherwise stated, the graph G has p vertices and qconsidering the existence of a path containing edges. Degree of a vertex v is denoted by d(v), theany three vertices of a graph G. In[3], G. maximum degree of a graph G is denoted by Δ(G).Mahadevan et. al., was introduced the concept of We denote a cycle on p vertices by Cp, a path on ptriple connected domination number of a graph. vertices by Pp, and a complete graph on p verticesIn this paper, we introduce a new domination by Kp. A graph G is connected if any two vertices ofparameter, called restrained triple connected G are connected by a path. A maximal connecteddomination number of a graph. A subset S of V subgraph of a graph G is called a component of G.of a nontrivial graph G is called a dominating set The number of components of G is denoted byof G if every vertex in V − S is adjacent to at least 𝜔(G). The complement 𝐺 of G is the graph withone vertex in S. The domination number (G) of vertex set V in which two vertices are adjacent ifG is the minimum cardinality taken over all and only if they are not adjacent in G. A tree is adominating sets in G. A subset S of V of a connected acyclic graph. A bipartite graph (ornontrivial graph G is called a restrained bigraph) is a graph whose vertex set can be divideddominating set of G if every vertex in V − S is into two disjoint sets V1 and V2 such that every edgeadjacent to at least one vertex in S as well as has one end in V1 and another end in V2. A completeanother vertex in V - S. The restrained bipartite graph is a bipartite graph where everydomination number r(G) of G is the minimum vertex of V1 is adjacent to every vertex in V2. Thecardinality taken over all restrained dominating complete bipartite graph with partitions of ordersets in G. A subset S of V of a nontrivial graph G |V1|=m and |V2|=n, is denoted by Km,n. A star,is said to be triple connected dominating set, if S denoted by K1,p-1 is a tree with one root vertex and pis a dominating set and the induced sub graph – 1 pendant vertices. A bistar, denoted by B(m, n) is<S> is triple connected. The minimum the graph obtained by joining the root vertices of thecardinality taken over all triple connected stars K1,m and K1,n. The friendship graph, denoteddominating sets is called the triple connected by Fn can be constructed by identifying n copies ofdomination number and is denoted by 𝜸tc. A the cycle C3 at a common vertex. A wheel graph,subset S of V of a nontrivial graph G is said to be denoted by Wp is a graph with p vertices, formed byrestrained triple connected dominating set, if S is a connecting a single vertex to all vertices of Cp-1. If Srestrained dominating set and the induced sub is a subset of V, then <S> denotes the vertexgraph <S> is triple connected. The minimum induced subgraph of G induced by S. The opencardinality taken over all restrained triple neighbourhood of a set S of vertices of a graph G,connected dominating sets is called the restrained denoted by N(S) is the set of all vertices adjacent totriple connected domination number and is some vertex in S and N(S) ∪ S is called the closeddenoted by 𝜸rtc. We determine this number for neighbourhood of S, denoted by N[S]. A cut –some standard graphs and obtain bounds for vertex (cut edge) of a graph G is a vertex (edge)general graph. Its relationship with other graph whose removal increases the number oftheoretical parameters are also investigated. components. A vertex cut, or separating set of a connected graph G is a set of vertices whoseKey words: Domination Number, Triple removal results in a disconnected. The connectivityconnected graph, Triple connected domination or vertex connectivity of a graph G, denoted by κ(G)number, Restrained Triple connected domination (where G is not complete) is the size of a smallestnumber. vertex cut. The chromatic number of a graph G,AMS (2010): 05C69 denoted by χ(G) is the smallest number of colors needed to colour all the vertices of a graph G in1. Introduction which adjacent vertices receive different colours. By a graph we mean a finite, simple, For any real number 𝑥, 𝑥 denotes the largestconnected and undirected graph G(V, E), where V integer less than or equal to 𝑥. A Nordhaus - Gaddum-type result is a (tight) lower or upper 225 | P a g e
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G. Mahadevan, Selvam Avadayappan, V. G. Bhagavathi Ammal, T. Subramanian / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.225-229bound on the sum or product of a parameter of a Example 1.3 Let v1, v2, v3, v4, be the vertices of K4.graph and its complement. Terms not defined here The graph K4(2P2, 2P2, 2P3, P3) is obtained from K4are used in the sense of [2]. by attaching 2 times a pendant vertex of P2 on v1, 2 A subset S of V is called a dominating set times a pendant vertex of P2 on v2, 2 times a pendantof G if every vertex in V − S is adjacent to at least vertex of P3 on v3 and 1 time a pendant vertex of P3one vertex in S. The domination number (G) of G on v4 and is shown in Figure 1.1.is the minimum cardinality taken over alldominating sets in G. A dominating set S of aconnected graph G is said to be a connecteddominating set of G if the induced sub graph <S> is v1 v2connected. The minimum cardinality taken over allconnected dominating sets is the connected v4 v3domination number and is denoted by c. A dominating set is said to be restraineddominating set if every vertex in V – S is adjacentto atleast one vertex in S as well as another vertex inV - S. The minimum cardinality taken over all Figure 1.1 : K4(2P2, 2P2, 2P3, P3)restrained dominating sets is called the restraineddomination number and is denoted by 𝛾r. 2. Restrained Triple connected domination Many authors have introduced different numbertypes of domination parameters by imposing Definition 2.1 A subset S of V of a nontrivial graphconditions on the dominating set [11, 12]. Recently, G is said to be a restrained triple connectedthe concept of triple connected graphs has been dominating set, if S is a restrained dominating setintroduced by J. Paulraj Joseph et. al.,[9] by and the induced subgraph <S> is triple connected.considering the existence of a path containing any The minimum cardinality taken over all restrainedthree vertices of G. They have studied the properties triple connected dominating sets is called theof triple connected graphs and established many restrained triple connected domination number of Gresults on them. A graph G is said to be triple and is denoted by 𝛾rtc(G). Any triple connected twoconnected if any three vertices lie on a path in G. dominating set with 𝛾rtc vertices is called a 𝛾rtc -All paths, cycles, complete graphs and wheels are set of G.some standard examples of triple connected graphs. Example 2.2 For the graphs G1, G2, G3 and G4, inIn[3], G. Mahadevan et. al., was introduced the Figure 2.1, the heavy dotted vertices forms theconcept of triple connected domination number of a restrained triple connected dominating sets.graph. A subset S of V of a nontrivial graph G is saidto be a triple connected dominating set, if S is adominating set and the induced subgraph <S> istriple connected. The minimum cardinality takenover all triple connected dominating sets is calledthe triple connected domination number of G and isdenoted by 𝛾tc(G). Any triple connected dominatingset with 𝛾tc vertices is called a 𝛾tc -set of G. In[4, 5,6], G. Mahadevan et. al., was introducedcomplementary triple connected dominationnumber, complementary perfect triple connecteddomination number and paired triple connecteddomination number of a graph and investigatednew results on them. In this paper, we use this idea to developthe concept of restrained triple connecteddominating set and restrained triple connecteddomination number of a graph.Theorem 1.1 [9] A tree T is triple connected if andonly if T ≅ Pp; p ≥ 3.Notation 1.2 Let G be a connected graph with m Figure 2.1 : Graph with 𝜸rtc = 3.vertices v1, v2, …., vm. The graph obtained from G by Observation 2.3 Restrained Triple connectedattaching n1 times a pendant vertex of 𝑃𝑙1 on the dominating set (rtcd set) does not exists for allvertex v1, n2 times a pendant vertex of 𝑃𝑙2 on the graphs and if exists, then 𝛾rtc(G) ≥ 3.vertex v2 and so on, is denoted by G(n1 𝑃𝑙1 , n2 𝑃𝑙2 ,n3 𝑃𝑙3 , …., nm 𝑃𝑙 𝑚 ) where ni, li ≥ 0 and 1 ≤ i ≤ m. 226 | P a g e
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G. Mahadevan, Selvam Avadayappan, V. G. Bhagavathi Ammal, T. Subramanian / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.225-229Example 2.4 For the graph G5, G6 in Figure 2.2, we Example 2.11 Consider the graph G8 and itscannot find any restrained triple connected spanning subgraph H8 and G9 and its spanningdominating set. subgraph H9 shown in Figure 2.4. v3 v1 v2 v3 v1 v2 G8: H8: v4 v6 v4 v5 v6 v5 v1 v2 v1 v2Figure 2.2 : Graphs with no rtcd setThroughout this paper we consider only connectedgraphs for which triple connected two dominatingset exists. G9: H9: v5 v5Observation 2.5 The complement of the restrainedtriple connected dominating set need not be arestrained triple connected dominating set. v3 v4 v3 v4Example 2.6 For the graph G7 in Figure 2.3, 𝑆 ={v1, v4, v2} forms a restrained triple connecteddominating set of G3. But the complement V – S = Figure 2.4{v3, v5} is not a restrained triple connected For the graph G8, S = {v1, v4, v2} is a restrained tripledominating set. connected dominating set and so 𝛾rtc(G4) = 3. For the spanning subgraph H8 of G8, S = {v1, v4, v2, v5} is v2 a restrained triple connected dominating set so that v1 𝛾rtc(H8) = 4. Hence 𝛾rtc(G8) < 𝛾rtc(H8). For the graph G9, S = {v1, v2, v3} is a restrained triple connected dominating set and so 𝛾rtc(G9) = 3. For the spanning G7: v4 subgraph H9 of G9, S = {v1, v2, v3} is a restrained v3 triple connected dominating set so that 𝛾rtc(H9) = v5 3.Hence 𝛾rtc(G9) = 𝛾rtc(H9). Theorem 2.12 For any connected graph G with p ≥Figure 2.3 : Graph in which V–S is not a rtcd set 5, we have 3 ≤ 𝛾rtc(G) ≤ p - 2 and the bounds areObservation 2.7 Every restrained triple connected sharp.dominating set is a dominating set but not Proof The lower and bounds follows fromconversely. Definition 2.1. For K6, the lower bound is attainedObservation 2.8 Every restrained triple connected and for C9 the upper bound is attained.dominating set is a connected dominating set but not Theorem 2.13 For any connected graph G with 5conversely. vertices, 𝛾rtc(G) = p - 2 if and only if G ≅ K5, C5,Exact value for some standard graphs: F2, K5 – e, K4(P2), C4(P2 ), C3(P3), C3(2P2 ) and any1) For any cycle of order p ≥ 5, one of the following graphs given in Figure 2.5. 𝛾rtc(Cp) = 𝑝 − 2.2) For any complete graph of order p≥ 5, 𝛾rtc(Kp) = 3.3) For any complete bipartite graph oforder p ≥ 5, 𝛾rtc(Km,n) = 3. (where m, n ≥ 2 and m + n = p ).Observation 2.9 If a spanning sub graph H of agraph G has a restrained triple connecteddominating set, then G also has a restrained tripleconnected dominating set.Observation 2.10 Let G be a connected graph andH be a spanning sub graph of G. If H has arestrained triple connected dominating set, then 𝛾rtc(G) ≤ 𝛾rtc(H) and the bound is sharp. 227 | P a g e
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G. Mahadevan, Selvam Avadayappan, V. G. Bhagavathi Ammal, T. Subramanian / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.225-229 x x x G1: y z G2: y z G3: y z u v u v u v x x x G4: y z z G5: y G6: y z u v u v u v x x x G7: y z G8: y z z G9: y u v u v u vFigure 2.5Proof Suppose G is isomorphic to K5, C5, F2, K5 – e, Theorem 2.16 Let G be a graph such that G and 𝐺K4(P2), C4(P2 ), C3(P3 ), C3(2P2) and any one of the have no isolates of order p ≥ 5. Thengiven graphs in Figure 2.5., then clearly 𝛾rtc(G) = p (i) 𝛾rtc(G) + 𝛾rtc(𝐺 ) ≤ 2p -4- 2. Conversely, Let G be a connected graph with 5 (ii) 𝛾rtc(G). 𝛾rtc(𝐺 ) ≤ (p – 2)2 and the bound isvertices, and 𝛾rtc(G) = 3. Let S = {x, y, z} be the sharp. 𝛾rtc(G) –set of G. Take V – S = {u, v} and hence <V Proof The bound directly follows from Theorem– S> = K2 = uv. 2.12. For cycle C5, both the bounds are attained. Case (i) <S> = P3 = xyz. Since G is connected, x (or equivalently z) 3 Relation with Other Graph Theoreticalis adjacent to u (or equivalently v) (or) y is adjacent Parametersto u (or equivalently v). If x is adjacent to u. Since S Theorem 3.1 For any connected graph G with p ≥ 5is a restrained triple connected dominating set, v is vertices, 𝛾rtc(G) + κ(G) ≤ 2p – 3 and the bound isadjacent to x (or) y (or) z. If v is adjacent to z, then sharp if and only if G ≅ K5.G ≅ C5. If v is adjacent to y, then G ≅ C4(P2). Now Proof Let G be a connected graph with p ≥ 5by increasing the degrees of the vertices of K2 = uv, vertices. We know that κ(G) ≤ p – 1 and bywe have G ≅ G1 to G5, K5 – e, C3(P3). Now let y be Theorem 2.12, 𝛾rtc(G) ≤ p – 2. Hence 𝛾rtc(G) + κ(G)adjacent to u. Since S is a restrained triple connected ≤ 2p – 3. Suppose G is isomorphic to K5. Thendominating set, v is adjacent to x (or) y (or) z. If v is clearly 𝛾rtc(G) + κ(G) = 2p – 3. Conversely, Letadjacent to y, then G ≅ C3(2P2). If v is adjacent to y 𝛾rtc(G) + κ(G) = 2p – 3. This is possible only ifand z, x is adjacent to z, then G ≅ K4(P2). Now by 𝛾rtc(G) = p – 2 and κ(G) = p – 1. But κ(G) = p – 1,increasing the degrees of the vertices, we have G ≅ and so G ≅ Kp for which 𝛾rtc(G) = 3 = p – 2 . HenceG6 to G8, C3(2P2). G ≅ K5.Case (ii) <S> = C3 = xyzx. Theorem 3.2 For any connected graph G with p ≥ 5 Since G is connected, there exists a vertex in vertices, γrtc(G) + χ(G) ≤ 2p – 2 and the bound isC3 say x is adjacent to u (or) v. Let x be adjacent to sharp if and only if G ≅ K5.u. Since S is a restrained triple connected Proof Let G be a connected graph with p ≥ 5dominating set, v is adjacent to x, then G ≅ F2. Now vertices. We know that 𝜒(G) ≤ p and by Theoremby increasing the degrees of the vertices, we have G 2.12, 𝛾rtc(G) ≤ p – 2. Hence 𝛾rtc(G) + 𝜒(G) ≤ 2p – 2.≅ G9, K5. In all the other cases, no new graph exists. Suppose G is isomorphic to K5. Then clearly 𝛾rtc(G)The Nordhaus – Gaddum type result is given below: + 𝜒(G) = 2p – 2. Conversely, let 𝛾rtc(G) + 𝜒(G) = 228 | P a g e
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G. Mahadevan, Selvam Avadayappan, V. G. Bhagavathi Ammal, T. Subramanian / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.225-2292p – 2. This is possible only if 𝛾rtc(G) = p – 2 and Ciencia Indica, Vol. XXXI M, No. 2.: 847– 𝜒(G) = p. Since 𝜒(G) = p, G is isomorphic to Kp for 853.which 𝛾rtc(G) = 3 = p – 2. Hence G ≅ K5. [11] Sampathkumar, E.; Walikar, HB (1979):Theorem 3.3 For any connected graph G with p ≥ 5 The connected domination number of avertices, 𝛾rtc(G) + ∆(G) ≤ 2p – 3 and the bound is graph, J. Math. Phys. Sci 13 (6): 607–613.sharp. [12] Teresa W. Haynes, Stephen T. HedetniemiProof Let G be a connected graph with p ≥ 5 and Peter J. Slater (1998): Domination invertices. We know that ∆(G) ≤ p – 1 and by graphs, Advanced Topics, Marcel Dekker,Theorem 2.12, 𝛾rtc(G) ≤ p. Hence 𝛾rtc(G) + ∆(G) ≤ New York.2p – 3. For K5, the bound is sharp. [13] Teresa W. Haynes, Stephen T. Hedetniemi and Peter J. Slater (1998): Fundamentals ofREFERENCES domination in graphs, Marcel Dekker, New [1] Cokayne E. J. and Hedetniemi S. T. (1980): York. Total domination in graphs, Networks, Vol.10: 211–219. [2] John Adrian Bondy, Murty U.S.R. (2009): Graph Theory, Springer, 2008. [3] Mahadevan G., Selvam A., Paulraj Joseph J., and Subramanian T. (2012): Triple connected domination number of a graph , IJMC, Vol.3. [4] Mahadevan G., Selvam A., Paulraj Joseph J., Ayisha B., and Subramanian T. (2012): Complementary triple connected domination number of a graph , Accepted for publication in Advances and Applications in Discrete Mathematics, ISSN 0974-1658. [5] Mahadevan G, Selvam Avadayappan, Mydeen bibi A., Subramanian T. (2012): Complementary perfect triple connected domination number of a graph, International Journal of Engineering Research and Application, ISSN 2248- 9622, Vol.2, Issue 5,Sep – Oct, pp 260- 265. [6] Mahadevan. G, Selvam Avadayappan, Nagarajan. A, Rajeswari. A, Subramanian. T. (2012): Paired Triple connected domination number of a graph, International Journal of Computational Engineering Research, Vol. 2, Issue 5, Sep. 2012, pp. 1333-1338. [7] Nordhaus E. A. and Gaddum J. W. (1956): On complementary graphs, Amer. Math. Monthly, 63: 175–177. [8] Paulraj Joseph J. and Arumugam. S. (1997): Domination and coloring in graphs, International Journal of Management Systems, 8 (1): 37–44. [9] Paulraj Joseph J., Angel Jebitha M.K., Chithra Devi P. and Sudhana G. (2012): Triple connected graphs, Indian Journal of Mathematics and Mathematical Sciences, ISSN 0973-3329 Vol. 8, No. 1: 61-75. [10] Paulraj Joseph J. and Mahadevan G. (2006): On complementary perfect domination number of a graph, Acta 229 | P a g e
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