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### Ar25260265

1. 1. G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.260-265Complementary Perfect Triple Connected Domination Number of a Graph G. Mahadevan*, Selvam Avadayappan**, A.Mydeen bibi***, T.Subramanian**** * Dept. of Mathematics, Anna University of Technology, Tirunelveli-627 002. ** Dept.of Mathematics, VHNSN College, Virudhunagar. *** Research Scholar, Mother Teresa Women’s University, Kodaikanal. **** Research Scholar, Anna University of Technology Tirunelveli, Tirunelveli.Abstract: The number of components of G is denoted by 𝜔 (G). The concept of triple connected graphs The complement 𝐺 of G is the graph with vertex setwith real life application was introduced in [9] by V in which two vertices are adjacent if and only ifconsidering the existence of a path containing any they are not adjacent in G. A tree is a connectedthree vertices of G. A graph G is said to be triple acyclic graph. A bipartite graph (or bigraph) is aconnected if any three vertices lie on a path in G. graph whose vertices can be divided into two disjointIn [3], the concept of triple connected dominating sets U and V such that every edge connects a vertexset was introduced. A set S V is a triple in U to one in V. A complete bipartite graph is aconnected dominating set if S is a dominating set special kind of bipartite graph where every vertex ofof G and the induced sub graph <S> is triple the first set is connected to every vertex of the secondconnected. The triple connected domination set. The complete bipartite graph with partitions ofnumber 𝛾 tc(G) is the minimum cardinality taken order |V1|=m and |V2|=n, is denoted Km,n. A star,over all triple connected dominating sets in G. In denoted by K1,p-1 is a tree with one root vertex and p –this paper, we introduce a new domination 1 pendant vertices. A bistar, denoted by B(m, n) isparameter, called Complementary perfect triple the graph obtained by joining the root vertices of theconnected domination number of a graph. A set stars K1,m and K1,n.S V is a complementary perfect triple connected The friendship graph, denoted by Fn can bedominating set if S is a triple connected constructed by identifying n copies of the cycle C3 atdominating set of G and the induced sub graph a common vertex. A wheel graph, denoted by Wp is a<V - S> has a perfect matching. The graph with p vertices, formed by connecting a singlecomplementary perfect triple connected vertex to all vertices of an (p-1) cycle. A helm graph, denoted by Hn is a graph obtained from the wheel Wndomination number 𝛾 cptc(G) is the minimum by joining a pendant vertex to each vertex in thecardinality taken over all complementary perfect outer cycle of Wn by means of an edge. Corona oftriple connected dominating sets in G. We two graphs G1 and G2, denoted by G1 G2 isdetermine this number for some standard classesof graphs and obtain some bounds for general the disjoint union of one copy of G1 and |V1| copies ofgraph. Their relationships with other graph G2 (|V1| is the number of vertices in G1) in which iththeoretical parameters are investigated. vertex of G1 is joined to every vertex in the ith copy of G2. For any real number , denotes the largestKey words: Domination Number, Triple connected integer less than or equal to . If S is a subset of V,graph, Complementary perfect triple connected then <S> denotes the vertex induced subgraph of Gdomination number. induced by S. The open neighbourhood of a set S ofAMS (2010): 05C 69 vertices of a graph G, denoted by N(S) is the set of all vertices adjacent to some vertex in S and N(S) S is1 Introduction By a graph we mean a finite, simple, called the closed neighbourhood of S, denoted byconnected and undirected graph G (V, E), where V N[S]. The diameter of a connected graph is thedenotes its vertex set and E its edge set. Unless maximum distance between two vertices in G and isotherwise stated, the graph G has p vertices and q denoted by diam (G). A cut – vertex (cut edge) of aedges. Degree of a vertex v is denoted by d (v), the graph G is a vertex (edge) whose removal increasesmaximum degree of a graph G is denoted by Δ(G). the number of components. A vertex cut, orWe denote a cycle on p vertices by Cp, a path on p separating set of a connected graph G is a set ofvertices by Pp, and a complete graph on p vertices by vertices whose removal renders G disconnected. TheKp. A graph G is connected if any two vertices of G connectivity or vertex connectivity of a graph G,are connected by a path. A maximal connected denoted by κ(G) (where G is not complete) is the sizesubgraph of a graph G is called a component of G. of a smallest vertex cut. A connected subgraph H of a 260 | P a g e
2. 2. G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.260-265connected graph G is called a H -cut if (G – H) ≥ Theorem 1.1 [9] A tree T is triple connected if and2. The chromatic number of a graph G, denoted by only if T Pp; p ≥ 3.χ(G) is the smallest number of colors needed to Theorem 1.2 [9] A connected graph G is not triplecolour all the vertices of a graph G in which adjacent connected if and only if there exists a H -cut withvertices receive different colour. Terms not defined (G – H) ≥ 3 such that = 1 for athere are used in the sense of [2]. least three components C1, C2, and C3 of G – H. A subset S of V is called a dominating set of Notation 1.3 Let G be a connected graph with mG if every vertex in V − S is adjacent to at least one vertices v1, v2, …., vm. The graph G(n1 , n2 ,vertex in S. The domination number (G) of G is theminimum cardinality taken over all dominating sets n3 , …., nm ) where ni, li ≥ 0 and 1 ≤ i ≤ m, isin G. A dominating set S of a connected graph G is obtained from G by attaching n1 times a pendantsaid to be a connected dominating set of G if theinduced sub graph <S> is connected. The minimum vertex of on the vertex v1, n2 times a pendantcardinality taken over all connected dominating set is vertex of on the vertex v2 and so on.the connected domination number and is denoted by Example 1.4 Let v1, v2, v3, v4, be the vertices of K4,c. A subset S of V of a nontrivial graph G is said to the graph K4 (2P2, P3, P4, P3) is obtained from K4 bybe Complementary perfect dominating set, if S is adominating set and the subgraph induced by <V-S> attaching 2 times a pendant vertex of P 2 on v1, 1 timehas a perfect matching. The minimum cardinality a pendant vertex of P3 on v2, 1 times a pendant vertex of P4 on v3 and 1 time a pendant vertex of P 3 on v4taken over all Complementary perfect dominating and is shown in Figure 1.1.sets is called the Complementary perfect dominationnumber and is denoted by cp. One can get a comprehensive survey ofresults on various types of domination number of a v1 v2graph in [11]. Many authors have introduced differenttypes of domination parameters by imposing v4 v3conditions on the dominating set [1,8]. Recently theconcept of triple connected graphs was introduced byPaulraj Joseph J. et. al.,[9] by considering the Figure 1.1: K4 (2P2, P3, P4, P3)existence of a path containing any three vertices of G.They have studied the properties of triple connectedgraph and established many results on them. A graph 2 Complementary Perfect Triple ConnectedG is said to be triple connected if any three vertices Domination Numberlie on a path in G. All paths and cycles, complete Definition 2.1 A set S V is a complementarygraphs and wheels are some standard examples of perfect triple connected dominating set if S is a tripletriple connected graphs. connected dominating set of G and the induced sub A set S V is a triple connected dominating graph <V - S> has a perfect matching. Theset if S is a dominating set of G and the induced sub complementary perfect triple connected dominationgraph <S> is triple connected. The triple connected number cptc(G) is the minimum cardinality takendomination number tc(G) is the minimum over all complementary perfect triple connectedcardinality taken over all triple connected dominating dominating sets in G. Any triple connectedsets in G. dominating set with cptc vertices is called a cptc -set A set S V is a complementary perfect of G.triple connected dominating set if S is a triple Example 2.2 For the graph G1 in Figure 2.1, S = {v1,connected dominating set of G and the induced sub v2, v5} forms a cptc -set of G1. Hence cptc (G1) = 3.graph <V - S> has a perfect matching. Thecomplementary perfect triple connected domination v1number cptc(G) is the minimum cardinality takenover all complementary perfect triple connecteddominating sets in G. G 1: v5 v2 In this paper we use this idea to develop theconcept of complementary perfect triple connected v3 v4dominating set and complementary perfect tripleconnected domination number of a graph. Figure 2.1 Observation 2.3 Complementary perfect triple connected dominating set does not exists for allPrevious Results graphs and if exists, then cptc ≥ 3. 261 | P a g e
3. 3. G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.260-265 Example 2.4 For the graph G2 in Figure 2.2, any v6 v1 minimum dominating set must contain the supports and any connected dominating set containing these supports is not Complementary perfect triple connected and hence cptc does not exists. K6: v5 v2 G 2: v4 v3 Figure 2.2 Figure 2.5 Remark 2.5 Throughout this paper we consider only Theorem 2.12 If the induced subgraph of each connected graphs for which complementary perfect connected dominating set of G has more than two triple connected dominating set exists. pendant vertices, then G does not contain a Observation 2.6 The complement of the Complementary perfect triple connected dominating complementary perfect triple connected dominating set. set need not be a complementary perfect triple Proof This theorem follows from Theorem 1.2. connected dominating set. Example 2.13 For the graph G6 in Figure 2.6, Example 2.7 For the graph G3 in Figure 2.3, S = {v6, v2, v3, v4} is a minimum connected = {v1, v2, v3} forms a Complementary perfect triple dominating set so that c (G6) = 4. Here we notice connected dominating set of G3. But the complement that the induced subgraph of S has three pendant V – S = {v4, v5, v6, v7} is not a Complementary perfect vertices and hence G does not have a Complementary triple connected dominating set. perfect triple connected dominating set. v7 v1 v3 v2 v6 G 6: v7 v4 v1 v2 v6 v3 v4 v5G 3: v5 Figure 2.6 Figure 2.3 Observation 2.8 Every Complementary perfect triple Complementary perfect triple connected connected dominating set is a dominating set but not domination number for some standard graphs are the converse. given below. Example 2.9 For the graph G4 in Figure 2.4, S = {v1} 1) For any cycle of order p ≥ 5, cptc (Cp) = p – 2. is a dominating set, but not a complementary perfect 2) For any complete bipartite graph of order p ≥ 5, triple connected dominating set. cptc (Km,n) = v1 (where m, n ≥ 2 and m + n = p. 3) For any complete graph of order p ≥ 5, G 4: v4 v2 cptc (Kp) = v3 4) For any wheel of order p ≥ 5, Figure 2.4 Observation 2.10 For any connected graph G, (G) cptc (Wp) = ≤ cp(G) ≤ cptc(G) and the inequalities are strict. 5) For any Fan graph of order p ≥ 5, Example 2.11 For K6 in Figure 2.5, (K6) = {v1} = 1, cp(K6) = {v1, v2}= 2 and cptc(K6) = {v1, v2, v3, v4}= cptc (Fp) = 4. Hence (G) ≤ cp(G) ≤ cptc(G). 6) For any Book graphs of order p ≥ 6, cptc (Bp) = 4. 7) For any Friendship graphs of order p ≥ 5, cptc (Fn)) = 3. Observation 2.14 If a spanning sub graph H of a graph G has a Complementary perfect triple 262 | P a g e
4. 4. G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.260-265connected dominating set, then G also has a Proof The lower and upper bounds trivially followscomplementary perfect triple connected dominating from Definition 2.1. For C5, the lower bound isset. attained and for C9 the upper bound is attained.Example 2.16 For any graph G and H in Figure 2.7, Theorem 2.20 For a connected graph G with 5S = {v1, v4, v5} is a complementary perfect triple vertices, cptc(G) = p – 2 if and only if G isconnected dominating set and so cptc (G) = 3. For the isomorphic to C5, W5, K5, K2,3, F2, K5 – {e}, K4(P2),spanning subgraph H, S = {v1, v4, v5} is a C4(P2), C3(P3), C3(2P2) or any one of the graphscomplementary perfect triple connected dominating shown in Figure 2.9.set and so cptc(H) = 3. H1: x H2: x V1 G: v y v y V5 V4 V2 z u z u H3: y H4: y V3 H: V1 x z x u u v z v V4 V5 V2 H5: x H6: x v y v y V3 u z u zFigure 2.7 H7: xObservation 2.17 Let G be a connected graph and Hbe a spanning sub graph of G. If H has acomplementary perfect triple connected dominating v yset, then cptc (G) ≤ cptc (H) and the bound is sharp.Example 2.18 For the graph G in Figure 2.8, u zS = {v1, v2, v7, v8} is a complementary perfect tripleconnected dominating set and so cptc (G) = 4. For the Figure 2.9spanning subgraph H of G, S = {v1, v2, v3, v6, v7, v8} Proof Suppose G is isomorphic to C5, W5, K5, K2,3,is a complementary perfect triple connected F2, K5 – {e}, K4(P2), C4(P2), C3(P3), C3(2P2) or anydominating set and so cptc (H) = 6. one of the graphs H1 to H7 given in Figure 2.9, then clearly cptc (G) = p – 2. Conversely, let G be a G: V1 V2 V3 V4 connected graph with 5 vertices and cptc (G) = 3. Let S = {x, y, z} be a cptc -set, then clearly <S> = P3 or C3. Let V – S = V (G) – V(S) = {u, v}, then <V – S> = K2 = uv. Case (i) <S> = P3 = xyz. Since G is connected, there exists a vertex say x (or y, V8 V7 V6 V5 z) in P3 is adjacent to u (or v) in K2, then cptc -set of G does not exists. But on increasing the degrees of the 7 V1 V2 V3 V4 vertices of S, let x be adjacent to u and z be adjacent H: to v. If d(x) = d(y) = d(z) = 2, then G  C5. Now by increasing the degrees of the vertices, by the above argument, we have G  K5, K5 – {e}, K4(P2), C4(P2), C3(P3) or any one of the graphs H1 to H7 given in Figure 2.9. Since G is connected, there exists a vertex say y in P3 is adjacent to u (or v) in K2, then cptc -set V8 V7 V6 V5 of G does not exists. But on increasing the degrees ofFigure 2.8 the vertices of S, let y be adjacent to v, x be adjacentTheorem 2.19 For any connected graph G with p ≥ 5, to u and v and z be adjacent to u and v. If d(x) = 3,we have 3  cptc(G) ≤ p - 2 and the bounds are sharp. 263 | P a g e
5. 5. G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.260-265d(y) = 3, d(z) = 3, then G  W5. Now by increasing 3. Complementary Perfect Triple Connectedthe degrees of the vertices, by the above argument, Domination Number and Other Graphwe have G  K2,3, C3(2P2). In all the other cases, no Theoretical Parametersnew graph exists. Theorem 3.1 For any connected graph G with p ≥ 5Case (ii) <S> = C3 = xyzx. vertices, cptc (G) + κ(G) ≤ 2p – 3 and the bound isSince G is connected, there exists a vertex say x (or y, sharp if and only if G  K5.z) in C3 is adjacent to u (or v) in K2, then S = {x, u, v} Proof Let G be a connected graph with p ≥ 5 vertices.forms a cptc -set of G so that cptc (G) = p – 2. If We know that κ(G) ≤ p – 1 and by Theorem 2.19, cptcd(x) = 3, d(y) = d(z) = 2, then G  C3(P3). If d(x) = 4, (G) ≤ p – 2. Hence cptc (G) + κ(G) ≤ 2p – 3. Supposed(y) = d(z) = 2, then G  F2. In all the other cases, no G is isomorphic to K5. Then clearly cptc (G) + κ (G)new graph exists. = 2p – 3. Conversely, Let cptc (G) + κ(G) = 2p – 3.Nordhaus – Gaddum Type result: This is possible only if cptc (G) = p – 2 and κ (G) =Theorem 2.21 Let G be a graph such that G and p – 1. But κ (G) = p – 1, and so G  Kp for which cptchave no isolates of order p ≥ 5, then (G) = 3 = p – 2 so that p = 5. Hence G  K5.(i)cptc (G) + cptc ( ) ≤ 2(p – 2) Theorem 3.2 For any connected graph G with p ≥ 5(ii) cptc (G) . cptc ( )≤ (p – 2)2 and the bounds are vertices, cptc (G) +  (G) ≤ 2p – 2 and the bound is sharp if and only if G  K5.sharp. Proof Let G be a connected graph with p ≥ 5 vertices.Proof The bounds directly follow from Theorem We know that (G) ≤ p and by Theorem 2.19,2.19. For the cycle C5, cptc (G) + cptc ( ) = 2(p – 2) cptc(G) ≤ p – 2. Hence cptc (G) + (G) ≤ 2p – 2.and cptc (G). cptc ( )≤ (p – 2)2. Suppose G is isomorphic to K5. Then clearlyTheorem 2.22 γcptc (G) ≥  p / ∆+1  cptc (G) + (G) = 2p – 2. Conversely, Let cptc (G)Proof Every vertex in V-S contributes one to degree + (G) = 2p – 2. This is possible only if cptc (G) =sum of vertices of S. Then |V - S|   u  S d(u) where p – 2 and (G) = p. But (G) = p, and so G isS is a complementary perfect triple connected isomorphic to Kp for which cptc (G) = 3 = p – 2 sodominating set. So |V - S|  γcptc ∆ which implies that p = 5. Hence G  K5.(|V|- |S|)  γcptc ∆. Therefore p - γcptc  γcptc ∆, which Theorem 3.3 For any connected graph G with p ≥ 5implies γcptc (∆+1) ≥ p. Hence γcptc (G) ≥  p / ∆+1  vertices, cptc (G) +(G) ≤ 2p – 3 and the bound isTheorem 2.23 Any complementary perfect triple sharp if and only if G is isomorphic to W5, K5, C3(2),connected dominating set of G must contains all the K5 – {e}, K4(P2), C3(2P2) or any one of the graphspendant vertices of G. shown in Figure 3.1.Proof Let S be any complementary perfect tripleconnected dominating set of G. Let v be a pendant G 1: v4 G 2: v4vertex with support say u. If v does not belong to S,then u must be in S, which is a contradiction S is acomplementary perfect triple connected dominating v3 v v3 vset of G. Since v is a pendant vertex, so v belongs toS. v2 v1 v2 v1Observation 2.24 There exists a graph for which G 3: v4 v4γcp (G) = γtc(G) = γcptc (G) is given below. G 4: v1 v2 v3 v v3 v v6 v2 v1 v2 v1 v5 Figure 3.1 Proof Let G be a connected graph with p ≥ 5 vertices. v4 We know that (G) ≤ p – 1 and by Theorem 2.19, cptc (G) ≤ p – 2. Hence cptc (G) + (G) ≤ 2p – 3. Let v3 v7 G be isomorphic to W5, K5,C3(2), K5 – {e}, K4(P2), C3(2P2) or any one of the graphs G1 to G4 given in H: Figure 3.1, then clearly cptc (G) +(G) = 2p – 3.Figure 2.10 Conversely, Let cptc (G) + (G) = 2p – 3. This isFor the graph Hi in figure 2.10, S = {v1, v3, v7} is a possible only if cptc (G) = p – 2 and (G) = p – 1.triple connected dominating set complementary But cptc (G) = p – 2 and (G) = p – 1, byperfect and complementary perfect triple connected Theorem 2.20, we have G  W5, K5, C3(2), K5 – {e},dominatin set. Hence γcp (G) = γtc(G) = γcptc (G) = 3. K4(P2), C3(2P2) and the graphs G1 to G4 given in Figure 3.1. 264 | P a g e
6. 6. G. Mahadevan, Selvam Avadayappan, A.Mydeen bibi, T.Subramanian / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.260-265REFERENCES [1] Cokayne E. J. and Hedetniemi S. T. (1980): Total domination in graphs, Networks, Vol.10: 211–219. [2] John Adrian Bondy, Murty U.S.R. (2009): Graph Theory [3] Mahadevan G. SelvamAvadayappan, Paulraj Joseph. J, Subramanian T: (2012) Tripleconnected domination number of a graph – Preprint [4] Nordhaus E. A. and Gaddum J. W. (1956): On complementary graphs, Amer. Math. Monthly, 63: 175–177. [5] Paulraj Joseph J. and Arumugam. S. (1992): Domination and connectivity in graphs, International Journal of Management Systems, 8 (3): 233–236. [6] Paulraj Joseph J. and Arumugam. S. (1995): Domination in graphs, International Journal of Management Systems, 11: 177–182. [7] Paulraj Joseph J. and Arumugam. S. (1997): Domination and coloring in graphs, International Journal of Management Systems, 8 (1): 37–44. [8] Paulraj Joseph J. , Mahadevan G. and Selvam A. (2006): On Complementary perfect domination number of a graph, Acta Ciencia Indica, Vol. XXXI M, No. 2, 847 – 854. [9] Paulraj Joseph J., Angel Jebitha M.K., Chithra Devi P. and Sudhana G. (2012): Triple connected graphs, (To appear) Indian Journal of Mathematics and Mathematical Sciences. [10] Sampathkumar, E.; Walikar, HB (1979): The connected domination number of a graph, J. Math. Phys. Sci 13 (6): 607–613. [11] Teresa W. Haynes, Stephen T. Hedetniemi and Peter J. Slater (1998): Domination in graphs, Advanced Topics, Marcel Dekker, New York. [12] Teresa W. Haynes, Stephen T. Hedetniemi and Peter J. Slater (1998): Fundamentals of domination in graphs, Marcel Dekker, New York. 265 | P a g e