International Journal of Advanced Scientific Research and Management, Volume 3 Issue 8, Aug 2018 www.ijasrm.com
ISSN 2455-6378
Triple Connected Line Domination Number For Some Standard And Special Graphs
A.Robina Tony1, Dr. A.Punitha Tharani2
1 Research Scholar (Register Number: 12519), Department of Mathematics, St.Mary’s College(Autonomous), Thoothukudi – 628 001, Tamil Nadu, India, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli – 627 012, Tamil Nadu, India
2 Associate Professor, Department of Mathematics, St.Mary’s College(Autonomous), Thoothukudi – 628 001, Tamil Nadu, India, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627 012, Tamil Nadu, India
Abstract vertex domination, the concept of edge The concept of triple connected graphs with real life domination was introduced by Mitchell and application was introduced by considering the Hedetniemi. A set F of edges in a graph G is existence of a path containing any three vertices of a called an edge dominating set if every edge in E graph G. A subset S of V of a non - trivial graph G is – F is adjacent to at least one edge in F. The said to be a triple connected dominating set, if S is a dominating set and the induced subgraph is edge domination number γ '(G) of G is the triple connected. The minimum cardinality taken minimum cardinality of an edge dominating set over all triple connected dominating set is called the of G. An edge dominating set F of a graph G is triple connected domination number and is denoted a connected edge dominating set if the edge by γtc(G). A subset D of E of a non- trivial graph G is induced subgraph 〈F〉 is connected. The said to be a triple connected line dominating set, if D connected edge domination number γ'c(G) of G is an edge dominating set and the edge induced is the minimum cardinality of a connected edge subgraph is triple connected. The minimum All graphs considered here are finite, undirected cardinality taken over all triple connected without loops and multiple edges. Unless and dominating sets is called the triple connected otherwise stated, the graph G = (V, E) domination number and is denoted by γtc. A considered here have p = vertices and q = subset D of E of a nontrivial connected graph G edges. A set S of vertices in a graph G is is said to be a triple connected line dominating called a dominating set if every vertex in V – S set, if D is an edge dominating set and the edge is adjacent to some vertex in S. The domination induced subgraph
International Journal of Advanced Scientific Research and Management, Volume 3 Issue 8, Aug 2018 www.ijasrm.com
ISSN 2455-6378
triple connected line domination number of G ’ ’ and is denoted by γtcl (G). γtcl (Cp) =
2. Exact value for some standard graphs Proof Let Cp be a cycle of order p ≥ 3 and let D be a
2.1. Proposition triple connected line dominating set of Cp
For any path Pp of order p ≥ 3, Case (i) p=3 Then
Case (ii) p≥ 4 The only connected subgraphs of any cycle are paths, which are triple connected. If Proof = 2, then D is an edge dominating set Let Pp be a path of order p ≥ 3 and let D and and
Case (i) p < 5 Hence, D is a triple connected line Since the existence of a triple connected dominating set. If > 2, then take any line dominating set of a connected graph G three consecutive edges x, y, z . The ’ edge y is not dominated by any edge in D since implies that γtcl (G) ≥ 2, we need to show the the only neighbors of y are x and z. Therefore, existence of such a set D. For P3 it is obvious. exactly two edges can be removed from Cp to For P4, take any two consecutive edges to be in D. Therefore, D is an edge dominating set and obtain a triple connected line dominating set of minimum cardinality.
Let Bm,n be a bistar of order p, where m 2.2. Proposition + n + 2 = p, m,n ≥ 1, and p ≥ 4. If the edge connecting the root vertices of the bistar is in D, For any cycle Cp of order p ≥ 3, then D is already an edge dominating set. Thus,
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International Journal of Advanced Scientific Research and Management, Volume 3 Issue 8, Aug 2018 www.ijasrm.com
ISSN 2455-6378
to obtain a triple connected line dominating set, we only need to add one more edge to D. ’ For the Truncated Tetrahedron G, γtcl (G) = ’ Hence,
3.1. The Franklin graph is a 3- regular graph 3.3. The Herschel graph is a bipartite with 12 vertices and 18 edges as shown in undirected graph with 11 vertices and 18 the Fig.1 edges as shown in Fig 3, the smallest non hamiltonian polyhedral graph. It is named after British astronomer Alexander Stewart Herschel
Fig. 1
’ For the Franklin graph G, γtcl (G) = 9 . Here D =
{e3, e4, e6, e8, e9, e12, e14, e16 , e18} is a minimum triple connected line domination number of a graph. 3.2. The Truncated Tetrahedron is an
Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and Fig. 3 18 edges as shown in Fig.2 Archimedean solid ’ means one of 13 possible solids whose faces are For the Herschel graph G, γtcl (G) = 7. Here all regular polygons whose polyhedral angles D = {e1, e2, e3, e4, e5, e6, e7} is a minimum are all equal. triple connected line domination number of a graph
3.4. The Golomb graph is a unit distance graph discovered around 1960 – 1965 by Golomb with 10 vertices and 18 edges as shown in the Fig 4
Fig. 2
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International Journal of Advanced Scientific Research and Management, Volume 3 Issue 8, Aug 2018 www.ijasrm.com
ISSN 2455-6378
4.Conclusion
In this paper we provided the triple connected line domination number of some special and standard graphs. Further, the bounds for general graph and its relationship with other graph theoretical parameters can also be investigated.
References Fig. 4 ’ [1] John Adrian Bondy, Murty For the Golomb graph G, γ (G) = 7 . Here tcl U.S.R.(2009): Graph Theory, D = {e1, e4, e7, e9, e10, e11, e12} is a minimum Springer, 2008. triple connected line domination number of [2] V.R.Kulli, College Graph a graph. Theory,Vishwa International Publications,Gulbarga, India (2012). 3.5 The Goldner–Harary graph is a simple [3] V.R.Kulli, Theory of undirected graph with 11 vertices and 27 Domination in Graphs, Vishwa edges as shown in Fig 5. It is named after A. International Publications, Gulbarga, Goldner and Frank Harary, who proved in India (2010). [4] V.R.Kulli and S.C.Sigarakanti, 1975 that it was the smallest non The connected edge domination hamiltonian maximal planar graph. number of a graph, Technical Report 8801, Dept Math. Gulbarga University, (1988). [5] 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. [6] S. L. Mitchell and S. T. Hedetniemi,Edge domination in trees. Congr. Numer.19 (1997) 489-
509. Fig. 5 [7] Paulraj Joseph J., Angel Jebitha M.K.,Chithra Devi P. and Sudhana ’ For the Goldner–Harary graph G, γtcl (G) = G. (2012):Triple connected graphs, 5. Here D = {e8, e15, e16, e17, e27} is a Indian Journal of Mathematics and minimum triple connected line domination Mathematical Sciences, ISSN 0973- number of a graph. 3329 Vol. 8,No. 1: 61-75. [8] Teresa W. Haynes, Stephen T. Hedetniemi and Peter J. Slater (1998):Fundamentals of domination in graphs,Marcel Dekker, New York.
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