International Journal of Mathematical Archive-4(10), 2013, 178-189 Available online through www.ijma.info ISSN 2229 – 5046
TRIPLE CONNECTED.COM DOMINATION NUMBER OF A GRAPH
G. Mahadevan1*, M. Lavanya2, Selvam Avadayappan3 and T. Subramanian4
1Dept. of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram - 624302.
Current Address: Dept. of Mathematics, Anna University: Tirunelveli Region, Tirunelveli – 627 007
2Dept. of Mathematics, Anna University: Tirunelveli Region, Tirunelveli – 627 007.
3Dept.of Mathematics, VHNSN College, Virudhunagar – 626 001.
4Research Scholar, Dept. of Maths., Anna University: Tirunelveli Region, Tirunelveli – 627 007.
(Received on: 04-09-13; Revised & Accepted on: 03-10-13)
ABSTRACT In this paper we introduce new domination parameter called triple connected.com domination number of a graph with real life applications. 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 subgraph
AMS (2010): 05C69.
1. INTRODUCTION
By a graph we mean a finite, simple, connected and undirected 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 by d(v), the maximum degree of a graph G is denoted by Δ(G). A graph G is connected if any two vertices of G are connected by a path. A maximal connected subgraph of a graph G is called a component of G. The number of components of G is denoted by (G). The complement 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 Cp, a path on p vertices by Pp, and a complete graph on ̅ p vertices by𝜔𝜔 Kp. A wheel graph Wn𝐺𝐺 of order n, sometimes simply called an n-wheel, is a graph that contains a cycle of order n-1, and for which every vertex in the cycle is connected to one other vertex. A tree is a connected acyclic graph. A bipartite graph (or bigraph) is a graph whose vertex set can be divided into two disjoint sets V1 and V2 such that every edge has one end in V1 and another end in V2. A complete bipartite graph is a bipartite graph where every vertex of V1 is adjacent to every vertex in V2. The complete bipartite graph with partitions of order |V1|=m and |V2|=n, is denoted by Km,n. A star, denoted by K1,p-1 is a tree with one root vertex and p – 1 pendant vertices. The friendship graph, denoted by Fn can be constructed by identifying n copies of the cycle C3 at a common vertex. A helm graph, denoted by Hn is a graph obtained from the wheel Wn by joining a pendant vertex to each vertex in the outer cycle of Wn by means of an edge. The -book graph Bm is defined as the graph Cartesian product , where is a star graph and is the path graph on two nodes.
A cut – vertex (cut edge) of a graph G 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. The connectivity or vertex connectivity of a graph G, denoted by κ(G) (where G is not complete) is the size of a smallest vertex cut. The chromatic number of a graph G, denoted by χ(G) is the smallest number of colours needed to colour all the vertices of a graph G in which adjacent vertices receive different colour. For any real number , denotes the largest integer less than or equal to . A Nordhaus -Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. Terms not defined here are used in the sense of𝑥𝑥 [15⌊𝑥𝑥].⌋ 𝑥𝑥Corresponding author: G. Mahadevan1* 1 Dept. of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram - 624302".
International Journal of Mathematical Archive- 4(10), Oct. – 2013 178 G. Mahadevan1*, M. Lavanya2, Selvam Avadayappan3 and T. Subramanian4/ Triple Connected.com Domination Number of a Graph/ IJMA- 4(10), Oct.-2013.
A subset S of V is called a dominating set of G if every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) of G is the minimum cardinality 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 induced sub graph is connected. The minimum cardinality taken over all connected dominating sets is the connected domination number and is denoted by γc.
Many authors have introduced different types of domination parameters by imposing conditions on the dominating set [19, 20, 21].
In [18] Paulraj Joseph et. al., introduced the concept of triple connected graphs.
In [3] Mahadevan et. al., introduced the concept of triple connected domination number of a graph.
A subset S of V of a nontrivial graph G is said to be an triple connected dominating set, if S is a dominating set and the induced sub graph 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 tc (G).
In [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] G. Mahadevan et. al., introduced𝛾𝛾 complementary triple connected domination number, complementary perfect triple connected domination number, paired triple connected domination number, triple connected two domination number, restrained triple connected domination number, dom strong triple connected domination number, strong triple connected domination number, weak triple connected domination number, triple connected complementary tree domination number of a graph, efficient complementary perfect triple connected domination number of a graph, efficient triple connected domination number of a graph respectively and investigated new results on them.
In this paper, we use this idea to develop the concept of triple connected.com dominating set and triple connected.com domination number of a graph.
Notation 1.1: [3] Let G be a connected graph with m vertices v1, v2, …., vm. The graph obtained from G by attaching n1 times a pendant vertex of 1 on the vertex v1, n2 times a pendant vertex of 2 on the vertex v2 and so on, is denoted by G(n1 , n2 , n3 , …., nm ) where ni, li ≥ 0 and 1 ≤ i ≤ m. 1 2 3 𝑃𝑃𝑙𝑙 𝑃𝑃𝑙𝑙 𝑙𝑙 𝑙𝑙 𝑙𝑙 𝑙𝑙𝑚𝑚 2. TRIPLE𝑃𝑃 𝑃𝑃 CONNECTED.COM𝑃𝑃 𝑃𝑃 DOMINATION NUMBER OF A GRAPH
Definition 2.1: A subset S of V of a nontrivial graph G is said to be an triple connected.com dominating set, if S is a triple connected dominating set and the induced subgraph
Example𝛾𝛾 2.2: For the graph G1 in figure 2.1, S = {v5, v6, v𝛾𝛾7} forms a tc.com -set of G𝛾𝛾1. Hence tc.com(G1) = 3.
𝛾𝛾 𝛾𝛾
Figure - 2.1: Graph with tc.com = 3.
Real Life Application of Triple Connected.com Domination Number𝛾𝛾
Today in the modern world, there may situations arises where police department needs help from fire services and ambulance service to rescue peoples from disasters. Therefore it is necessary to connect these three departments together for fast rescue the peoples from any disaster. Suppose there are collection of major cities which dominates some set of sub cities are connected by means of roadways. Suppose we have planned to locate police stations, fire service and ambulance service to the collection of major cities, it is necessary that each major cities in the collection should connect with any other two major cities in the collection, so that one department can contact and ask help from other department easily. Also sub cities itself are connected by means of roadways, then the rescuer can move from one sub city to another. If we draw a graph of the above situation by taking all the given cities as its vertices and the roadways as its edges, then the solution is nothing but the finding the triple connected.com dominating set of the graph. © 2013, IJMA. All Rights Reserved 179 G. Mahadevan1*, M. Lavanya2, Selvam Avadayappan3 and T. Subramanian4/ Triple Connected.com Domination Number of a Graph/ IJMA- 4(10), Oct.-2013.
Thus the above problem of locating the police stations, fire and ambulance services in to major cities is reduces in to the triple connected.com dominating set of the constructed graph.
Observation 2.3: Triple connected.com dominating set ( tc.com -set or tc.com set) does not exists for all graphs and if exists, tc.com(G) ≥ 3. 𝛾𝛾 Example𝛾𝛾 2.4: For K1,n, there does not exists any triple connected.com dominating set for p ≥ 5.
Remark 2.5: Throughout this paper we consider only connected graphs for which triple connected.com dominating set exists.
Observation 2.6: The complement of a triple connected.com dominating set need not be a triple connected.com dominating set.
Example 2.7: For the graph G2 in figure 2.2, = {v1, v4, v7} forms a triple connected.com dominating set of G2. But the complement V – S = {v2, v3, v5, v6} is not a triple connected.com dominating set. 𝑆𝑆
Figure - 2.2: Graph in which V – S is not a tc.com set
Observation 2.8: Every triple connected.com dominating set is a dominating set but not conversely.
Example 2.9: For the graph G3 in figure 2.3, = {v1, v2, v3} is a minimum triple connected.com dominating set but S1 = {v2, v5} is a dominating set of G3. 𝑆𝑆
Figure - 2.3
Observation 2.10: Every triple connected.com dominating set is a triple connected dominating set but not conversely.
Example 2.11: For the graph G4 in the figure 2.4, S = {v4, v5, v6} is a triple connected dominating set but not a triple connected.com dominating set.
Figure - 2.4
© 2013, IJMA. All Rights Reserved 180 G. Mahadevan1*, M. Lavanya2, Selvam Avadayappan3 and T. Subramanian4/ Triple Connected.com Domination Number of a Graph/ IJMA- 4(10), Oct.-2013.
Exact value for some standard graphs:
1) For any path of order p ≥ 4, tc.com(Pp) = p – 1. 3 = 4 2) For any cycle, tc.com(Cp) = 𝛾𝛾 2 > 4. 𝑖𝑖𝑖𝑖 𝑝𝑝 3) For any complete graph of order p ≥ 4, tc.com(K p ) = 3. 𝛾𝛾 � 4) For the wheel graph of order𝑝𝑝 p −≥ 4, tc.com𝑖𝑖𝑖𝑖 𝑝𝑝(Wp ) = 3. 𝛾𝛾
5) For the complete bipartite graph of order𝛾𝛾 p ≥ 4, tc.com(Km,n ) = 3, (where m + n = p and m, n ≥ 2).
6) For the star graph of order p = 4, tc.com(K1,3 ) = 3.
7) For the friendship graph of order p = 5, tc.com(F𝛾𝛾2 ) = 3. 𝛾𝛾 Exact value for some named and special graphs:𝛾𝛾
1) The Bidiakis cube is a 3-regular graph with 12 vertices and 18 edges as shown in figure 2.5.
Figure - 2.5
For the Bidiakis cube graph G, tc.com(G) = 6. Here S = {v1, v2, v3, v7, v9, v11} is a minimum triple connected.com dominating set.
2) The Franklin graph a 3-regular graph with 12 vertices and 18 edges as shown below in figure 2.6.
Figure -2.6
For the Franklin graph G, tc.com(G) = 6. Here S = {v1, v2, v3, v4, v5, v6} is a minimum triple connected.com dominating set.
3) The Frucht graph is a 3-regular graph with 12 vertices, 18 edges, and no nontrivial symmetries as shown below in figure 2.7.
© 2013, IJMA. All Rights Reserved 181 G. Mahadevan1*, M. Lavanya2, Selvam Avadayappan3 and T. Subramanian4/ Triple Connected.com Domination Number of a Graph/ IJMA- 4(10), Oct.-2013.
Figure -2.7
For the Frucht graph G, γtc.comG) = 6. Here S = {v3, v4, v8, v9, v10, v11} is a minimum triple connected.com dominating set.
4) The Wagner graph is a 3-regular graph with 8 vertices and 12 edges, as shown in figure 2.8, named after Klaus Wagner. It is the 8-vertex Mobius ladder graph. Mobius ladder is a cubic circulant graph with an even number ‘n’ vertices, formed from an n- cycle by adding edges connecting opposite pairs of vertices in the cycle.
Figure -2.8
For the Wagner graph G, tc.com(G) = 3. Here S = {v6, v7, v8} is a minimum triple connected.com dominating set.
5) The Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges as shown in figure 2.9, the smallest non hamiltonian polyhedral graph. It is named after British astronomer Alexander Stewart Herschel.
Figure -2.9
© 2013, IJMA. All Rights Reserved 182 G. Mahadevan1*, M. Lavanya2, Selvam Avadayappan3 and T. Subramanian4/ Triple Connected.com Domination Number of a Graph/ IJMA- 4(10), Oct.-2013.
For the Herschel graph G, γtc.com(G) = 5. Here S = {v1, v6, v8, v10, v11} is a minimum triple connected.com dominating set.
6) The Moser spindle (also called the Mosers' spindle or Moser graph) is an undirected graph, named after mathematicians Leo Moser and his brother William, with seven vertices and eleven edges as shown in figure 2.10.
Figure -2.10
For the Moser spindle graph G, γtc.com(G) = 3. Here S = {v1, v2, v3} is a minimum triple connected.com dominating set.
7) The Goldner–Harary graph is a simple undirected graph with 11 vertices and 27 edges as shown in figure 2.11. It is named after A. Goldner and Frank Harary, who proved in 1975 that it was the smallest non hamiltonian maximal planar graph.
Figure -2.11
For any Goldner- Harary graph, γtc.com(G) = 3. Here S = {v1, v4, v3} is a minimum triple connected.com dominating set.
8) The Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5 as shown in figure 2.12. It is named after German mathematician Herbert Grötzsch.
Figure -2.12 © 2013, IJMA. All Rights Reserved 183 G. Mahadevan1*, M. Lavanya2, Selvam Avadayappan3 and T. Subramanian4/ Triple Connected.com Domination Number of a Graph/ IJMA- 4(10), Oct.-2013.
For the Grötzsch graph, γtc.com(G) = 4. Here S = {v1, v2, v10, v11} is a minimum triple connected.com dominating set.
9) The Hoffman graph is a 4-regular graph with 16 vertices and 32 edges as shown in figure 2.13 discovered by Alan Hoffman.
Figure -2.13
For the Hoffman graph, γtc.com(G) = 6. Here S = {v1, v2, v6, v8, v15, v16} is a minimum triple connected.com dominating set.
10) The Möbius–Kantor graph is a symmetric bipartite cubic graph with 16 vertices and 24 edges as shown in figure 2.14, named after August Ferdinand Möbius and Seligmann Kantor.
Figure - 2.14
For the Mobius – Kantor graph, γtc.com(G) = 8. Here S = {v1, v2, v3, v4, v5, v6, v7, v8} is a minimum triple connected.com dominating set.
11) The Truncated Tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges as shown in figure 2.15. Archimedean solid means one of 13 possible solids whose faces are all regular polygons whose polyhedral angles are all equal.
© 2013, IJMA. All Rights Reserved 184 G. Mahadevan1*, M. Lavanya2, Selvam Avadayappan3 and T. Subramanian4/ Triple Connected.com Domination Number of a Graph/ IJMA- 4(10), Oct.-2013.
Figure - 2.15
For the Truncated Tetrahedron, γtc.com(G) = 6. Here S = {v2, v5, v8, v9, v10, v11} is a minimum triple connected.com dominating set.
12) The Desargues graph is a distance-transitive cubic graph with 20 vertices and 30 edges as shown in figure 2.16. It is named after Gerard Desargues.
Figure -2.16
For any Desargues graph, γtc.com(G) = 10. Here S = {v1, v2, v3, v4, v5, v6, v7, v8, v9, v10} is a minimum triple connected.com dominating set.
Remark 2.12: Triple connected.com domination number does not exists for the following graphs. 1) Helm graph 2) Star graph (if the order p > 4) 3) Friendship graph (if the order p > 5).
Theorem 2.13: For any connected graph G with p ≥ 4 vertices, we have 3 tc.com(G) ≤ p - 1 and the bounds are sharp.
Proof: Since any triple connected.com dominating set has at least 3 elements≤ 𝛾𝛾 and the complement V – S is connected so it contains at least one components. The theorem follows. For C5, the lower bound is attained and for P8 the upper bound is attained.
© 2013, IJMA. All Rights Reserved 185 G. Mahadevan1*, M. Lavanya2, Selvam Avadayappan3 and T. Subramanian4/ Triple Connected.com Domination Number of a Graph/ IJMA- 4(10), Oct.-2013.
Theorem 2.14: For any connected graph G with 4 vertices, tc.com(G) = p - 1 if and only if G is isomorphic to K4, P4, C4, W4, K1,3, K4- {e}, K3(P2). 𝛾𝛾 Proof: Suppose G is isomorphic to K4, P4, C4, W4, K1,3, K4- {e}, K3(P2), then clearly, tc.com(G) = p – 1. Conversely, assume that G is a connected graph with 4 vertices and, tc.com(G) = 3 = p – 1. Let S = {u, v, w} be a tc.com(G)-set. Then = P3 or C3. Let V – S = V(G) – V(S) = {x}. 𝛾𝛾 𝛾𝛾 𝛾𝛾 Case – 1: = P3 = uvw. Since G is connected, there exists a vertex u (or w) in P3 which is adjacent to x, then G P4. If u is adjacent to x and w is adjacent to x then G C4. If v is adjacent to x, then G K1,3. If u is adjacent to x, v is adjacent to x, then G K3(P2). If u is adjacent to x, v is adjacent to x, and w is adjacent to x, then G K4 – {e}≅. In all the other cases, no graph exists. ≅ ≅ ≅ ≅ Case - 2: = C3 = uvwu. Since G is connected, without loss of generality x is adjacent to u (say), then G K3(P2). If x is adjacent to u and v, then G K4 – {e}. If x is adjacent to u, v and w, then G W4. In all the other cases, no new graph exists. ≅ Theorem≅ 2.15: Let v be a cut vertex of a graph G and S≅ be a tc.com(G) – set of G. If v S, then all vertices except one component of G – v belongs to S. 𝛾𝛾 ∈ Proof: Suppose there exists vertices u and w in two different components of G – v and not is S. Since v is on every u – w path of G and v S, then the vertices u and w are not connected in
Theorem 2.16: Let∈ S be a tc.com(G) – set of G such that V – S is a dominating set. Then S has no cut vertices.
Proof: Suppose S has a cut𝛾𝛾 vertex say x, then by theorem 2.15, all the vertices except one of the components of G – {x} are in S. Hence V – S cannot dominate all the vertices of S.
The Nordhaus – Gaddum type result is given below:
Theorem 2.17: Let G be a graph such that G and have no isolates of order p ≥ 4. Then (i) tc.com(G) + tc.com( ) ≤ 2p - 2. 2 ̅ (ii) tc.com(G). tc.com( ) ≤ (p – 1) and the bound 𝐺𝐺is sharp. 𝛾𝛾 𝛾𝛾 𝐺𝐺̅ ̅ Proof𝛾𝛾: The bound𝛾𝛾 directly𝐺𝐺 follows from theorem 2.13. For P4, both the bounds are attained.
3. RELATION WITH OTHER GRAPH THEORETICAL PARAMETERS
Theorem 3.1: For any connected graph G with p ≥ 4 vertices, tc.com(G) + κ(G) ≤ 2p – 2 and the bound is sharp if and only if G K4. 𝛾𝛾 Proof: Let≅ G be a connected graph with p ≥ 4 vertices. We know that κ(G) ≤ p – 1 and by theorem 2.13, tc.com(G) ≤ p – 1. Hence tc.com(G) + κ(G) ≤ 2p – 2. Suppose G is isomorphic to K4, then clearly tc.com(G) + κ(G) ≤ 2p – 2. Let tc.com(G) + κ(G) ≤ 2p – 2. This is possible only if tc.com(G) = p - 1 and κ(G) = p – 1. But κ(G) = p – 1, G𝛾𝛾 Kp and for Kp, p ≥ 4, 𝛾𝛾tc.com(G) = 3 so that p = 4. Hence G K4. 𝛾𝛾 𝛾𝛾 𝛾𝛾 ≅ Theorem 𝛾𝛾3.2: For any connected graph G with ≅p ≥ 4 vertices, tc.com(G) + (G) ≤ 2p – 2 and the bound is sharp.
Proof: Let G be a connected graph with p ≥ 4 vertices. We know𝛾𝛾 that (G)∆ ≤ p – 1 and by theorem 2.13, tc.com(G) ≤ p - 1. Hence tc.com(G) + (G) ≤ 2p – 2. For K4, the bound is sharp. ∆ 𝛾𝛾 𝛾𝛾 ∆ Theorem 3.3: For any connected graph G with p ≥ 4 vertices, γtc.com(G) + χ(G) ≤ 2p – 1 and the bound is sharp if and only if G K4.
Proof: Let≅ G be a connected graph with p ≥ 4 vertices. We know that χ(G) ≤ p and by theorem 2.13, tc.com(G) ≤ p – 1. Hence tc.com(G) + χ(G) ≤ 2p – 1. Suppose G is isomorphic to K4, then clearly γtc.com(G) + χ(G) ≤ 2p – 1. Let γtc.com(G) + χ(G) ≤ 2p – 1. This is possible only if tc.com(G) = p - 1 and (G) = p. Since (G) = p, G is isomorphic𝛾𝛾 to Kp for which 𝛾𝛾tc.com(G) = 3 for p ≥ 4. Hence p = 4 and so G K4. 𝛾𝛾 𝜒𝜒 𝜒𝜒 The authors𝛾𝛾 are working for the complete characterization≅ of graphs for which the sum of c.com set and chromatic number, tc.com set and connectivity, tc.com and maximum degree for higher order which will be reported later. © 2013, IJMA. All Rights Reserved 186 G. Mahadevan1*, M. Lavanya2, Selvam Avadayappan3 and T. Subramanian4/ Triple Connected.com Domination Number of a Graph/ IJMA- 4(10), Oct.-2013.
4. RELATION WITH OTHER DOMINATION PARAMETERS
Observation 4.1: For any connected graph G with p ≥ 5 vertices, c(G) ≤ tc(G) ≤ tc.com(G) and the bound is sharp.
Example: For the graph H in figure 3.1, S = {v4, v5, v6} is the connected𝛾𝛾 dominating𝛾𝛾 𝛾𝛾 set as well as triple connected and triple connected.com dominating set and hence the bound is sharp.
Figure - 3.1
Theorem 4.2: For any connected graph G with p ≥ 5 vertices, tc.com(G) + tc(G) ≤ 2p – 3 and the bound is sharp.
Proof: Let G be a connected graph with p ≥ 5 vertices. We know𝛾𝛾 that tc(G)𝛾𝛾 ≤ p – 2 and by theorem 2.13, tc.com(G) ≤ p -1. Hence tc.com(G) + tc(G) ≤ 2p – 3. For P9, the bound is sharp. 𝛾𝛾 𝛾𝛾 Theorem 4.𝛾𝛾 3: For any𝛾𝛾 connected graph G with p ≥ 5 vertices, tc.com(G) + ptc(G) ≤ 2p – 2 and the bound is sharp.
Proof: Let G be a connected graph with p ≥ 5 vertices. We know𝛾𝛾 that ptc(G)𝛾𝛾 ≤ p – 1 and by theorem 2.13, tc.com(G) ≤ p -1. Hence tc.com(G) + ptc(G) ≤ 2p – 2. For P7, the bound is sharp. 𝛾𝛾 𝛾𝛾 Theorem 4.𝛾𝛾 4: For any𝛾𝛾 connected graph G with p ≥ 4 vertices, tc.com(G) + stc(G) ≤ 2p – 2 and the bound is sharp.
Proof: Let G be a connected graph with p ≥ 4 vertices. We know𝛾𝛾 that stc(G)𝛾𝛾 ≤ p – 1 and by theorem 2.13, tc.com(G) ≤ p - 1. Hence tc.com(G) + stc(G) ≤ 2p – 2. For P4, the bound is sharp. 𝛾𝛾 𝛾𝛾 5. CUBIC𝛾𝛾 GRAPHS𝛾𝛾 WITH EQUAL TRIPLE CONNECTED.COM DOMINATION NUMBER AND CHROMATIC NUMBER
Cubic graph of order 8
Theorem 5.1: Let G be a connected cubic graph on 8 vertices. Then tc.com = = 3 if and only if G is isomorphic to anyone of the graphs given in Figure 5.1. 𝛾𝛾
Figure - 5.1
Proof: Let S = {u, v, w} be a minimum triple connected.com dominating set of G and V – S = {x1, x2, x3, x4, x5}. Then the induced subgraph has the following possible cases. = K3, 3, K2 K1, P3. Since S is a triple connected.com dominating set and G is cubic, clearly ≠ K , , K K . The only possible case is = P . 3 3 2 1 𝐾𝐾� 3
𝐾𝐾� Without loss of generality, let v be adjacent to u and w. Let u be adjacent to x1, x2 and w be adjacent to x4 and x5. Since S is a dominating set, v must be adjacent to x3. Now x3 is adjacent to x1 and x2 (or x4 and x5) or x3 is adjacent to x1 (or x2) and x4 (or x5). If x3 is adjacent to x1 and x2, then x4 is adjacent to x1 (or x2) and x5, then x5 is adjacent to x2, so that G G1.
≅© 2013, IJMA. All Rights Reserved 187 G. Mahadevan1*, M. Lavanya2, Selvam Avadayappan3 and T. Subramanian4/ Triple Connected.com Domination Number of a Graph/ IJMA- 4(10), Oct.-2013.
If x3 is adjacent to x1 and x4. Since G is cubic x5 is adjacent to x1 and x2 or x4 and x2. If x5 is adjacent to x1 and x2, then x2 is adjacent to x4, so that G G2. If x5 is adjacent to x4 and x2 then x1 is adjacent to x2, so that G G3. In all the other cases, no graph exists. ≅ ≅ Cubic graph of order 10
Theorem 5.12: Let G be a connected cubic graph on 10 vertices. Then there exists no graph for which tc.com = = 3.
𝛾𝛾 Proof: By contradiction, let S = {u, v, w} be a minimum triple connected.com dominating set of G and V – S = {x1, x2, x3, x4, x5, x6, x7}. Then the induced subgraph has the following possible cases. = K3, 3, K2 K1, P3. Since S is a triple connected.com dominating set and G is cubic, clearly ≠ K , , K K and P . Here we notice that 3 3 2 1 𝐾𝐾� 3 ≠ 3 for any cubic graph of order 10. Hence no cubic graph of order 10 exists for which = = 3. tc.com 𝐾𝐾� tc.com
Cubic𝛾𝛾 graph of order 12 𝛾𝛾
Theorem 5.13: Let G be a connected cubic graph on 12 vertices. Then there exists no graph for which tc.com = = 3.
𝛾𝛾 Proof By contradiction, let S = {u, v, w} be a minimum triple connected.com dominating set of G and V – S = {x1, x2, x3, x4, x5, x6, x7, x8, x9}. Then the induced subgraph has the following possible cases. = K3, 3, K2 K1, P3. Since S is a triple connected.com dominating set and G is cubic, clearly ≠ K , , K K and P . Here we notice 3 3 2 1 3𝐾𝐾� ≠ 3 for any cubic graph of order 12. Hence no cubic graph of order 12 exists for which = = 3. tc.com 𝐾𝐾� tc.com
R𝛾𝛾 EFERENCES 𝛾𝛾
[1] E. Sampathkumar and, H. B. Walikar, The connected domination number of a graph, J.Math. Phys. Sci., 13 (6) (1979), 607–613.
[2] G. Mahadevan, A. Selvam, M. Hajmeeral, On connected efficient domination number and chromatic number of a graph -I, International Journal of Intelligent Information Processing, Vol. 2(2), July-Dec. (2008), 89-96.
[3] 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), 93-104.
[4] 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(1) (2013), 39-54.
[5] 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) , 260-265.
[6] 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), 1333-1338.
[7] G. Mahadevan, A. Selvam, B. Ayisha, and, T. Subramanian, Triple connected two domination number of a graph, International Journal of Computational Engineering Research Vol. 2, Issue 6 (2012),101-104.
[8] G. Mahadevan, A. Selvam, V. G. Bhagavathi Ammal and, T. Subramanian, Restrained triple connected domination number of a graph, International Journal of Engineering Research and Application, Vol. 2, Issue 6 (2012), 225-229.
[9] G. Mahadevan, A. Selvam, M. Hajmeeral and, T. Subramanian, Dom strong triple connected domination number of a graph, American Journal of Mathematics and Mathematical Sciences, Vol. 1, Issue 2 (2012), 29-37.
[10] G. Mahadevan, A. Selvam, V. G. Bhagavathi Ammal and, T. Subramanian, Strong triple connected domination number of a graph, International Journal of Computational Engineering Research, Vol. 3, Issue 1 (2013), 242-247.
[11] G. Mahadevan, A. Selvam, V. G. Bhagavathi Ammal and, T. Subramanian, Weak triple connected domination number of a graph, International Journal of Modern Engineering Research, Vol. 3, Issue 1 (2013), 342-345.
© 2013, IJMA. All Rights Reserved 188 G. Mahadevan1*, M. Lavanya2, Selvam Avadayappan3 and T. Subramanian4/ Triple Connected.com Domination Number of a Graph/ IJMA- 4(10), Oct.-2013.
[12] G. Mahadevan, A. Selvam, N. Ramesh and, T. Subramanian, Triple connected complementary tree domination number of a graph, International Mathematical Forum, Vol. 8, No. 14 (2013), 659-670.
[13] G. Mahadevan, B. Anitha, A. Selvam, and, T. Subramanian, Efficient complementary perfect triple connected domination number of a graph, accepted for publication in Journal of Ultra Scientist of Physical Sciences, Vol. 25, No. 2(2013), 257-268. 4).
[14] G. Mahadevan, N. Ramesh, A. Selvam, and, T. Subramanian, Efficient triple connected domination number of a graph, International Journal of Computational Engineering Research, Vol. 3, Issue 6 (2013), 1-6.
[15] J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, 2008.
[16] J. Paulraj Joseph and, S. Arumugam, Domination and connectivity in graphs, International Journal of Management Systems, 8 (3) (1992), 233–236.
[17] J. Paulraj Joseph and, S. Arumugam, Domination and coloring in graphs, International Journal of Management Systems, 8 (1) (1997), 37–44.
[18] 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.1 (2012), 61-75.
[19] J. Paulraj Joseph and, G. Mahadevan, On complementary perfect domination number of a graph, Acta Ciencia Indica, Vol. XXXI M, No. 2. (2006), 847–853.
[20] T. W. Haynes, S. T. Hedetniemi and, P. J. Slater, Domination in graphs, Advanced Topics, Marcel Dekker, New York (1998).
[21] T. W. Haynes, S. T. Hedetniemi and, P. J. Slater, Fundamentals of domination in graphs, Marcel Dekker, New York (1998).
[22] V. R. Kulli, B. Janakiram, The non split domination number of a graph, Indian J. Pure and Applied Math., 31(5) (2000), 545-550.
Source of support: Nil, Conflict of interest: None Declared
© 2013, IJMA. All Rights Reserved 189