International Journal of Pure and Applied Mathematics Volume 119 No. 12 2018, 2397-2403 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu
DOMINATION PARAMETERS OF LINE GRAPH OF SUNLET GRAPH
Dr.A.Shobana Dept. of Science & Humanities Sri Krishna College of Engineering and Technology Coimbatore, Tamilnadu, India [email protected]
Ms.B.Logapriya Dept. of Science & Humanities Sri Krishna College of Engineering and Technology Coimbatore, Tamilnadu, India [email protected]
Abstract
The line graph of an undirected graph G is another graph 퐋(퐆) that represents the adjacencies between edges of G. A set S in V is a dominating set of a graph G, if every vertex in V- S is adjacent to atleast one vertex in S. The minimum cardinality of dominating set in G is a domination number denoted by 후(퐆). In this paper, the domination number and split domination number of line graph of n-Sunlet graph are determined.
Keywords: Sunlet graph, Line graph, Dominating set, Domination number.
I. Introduction
The concept of domination was introduced by De Jaenisch in 1862 while studying the problems of identifying the minimum number of queens to dominate chessboard. But the fastest growth in study of dominating set in graph theory began in 1960. Later the concept of domination set and domination number was used by Ore in 1962, Cockayne and Hedetniemi in 1977.Berge defined the concept of domination number of graph. The Study of domination in graph theory is fastest growing area and it came as a result of study of games such as game of chess where the goal is to dominate various squares of a chessboard by certain chess pieces.
Domination in graphs has application to several fields. The concept of domination helps in finding the shortest route and longest route. Also used in School bus Routing Locating radar station problem, Computer communication networks, land surveying, Nuclear power plants problem, Modeling biological networks, modeling social networks, Electrical networks, Modeling problems,
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Coding theory. Line graphs connect many important areas in graph theory. It is used in identifying a maximum matching in a graph.
In this paper, the domination number of line graph of sunlet graph and Split domination number of n-sunlet graph have been determined. The neighborhood of vertices is considered in measuring the domination number. From the definition of domination, every vertex of a graph must be protected by its neighborhood.
Throughout this paper, we consider undirected graph without self loops and parallel edges.
II Preliminaries
Definition 2.1:
Let G= (V,E) be an undirected graph with the set of vertices V and set of edges E. It is also denoted as V(G) and E(G).
Definition 2.2:
The number of edges incident at the vertex 푣푖 is called the degree of the vertex with self loops counted twice and it is denoted by 푑(푣푖).
Definition 2.3:
If there is an edge from a vertex to itself then the edge is called Self loop. A graph H is said to be a sub graph of G if all the vertices and all the edges of H are in G and if the adjacency is preserved in H exactly as in G.
Definition 2.4:
The line graph 퐿(퐺) of a graph G has its vertices as the edges of G and two vertices of 퐿(퐺) are adjacent if the corresponding edges of G are adjacent.
Definition 2.5:
The n- sunlet graph is a graph on 2n vertices is obtained by attaching n -
pendant edges to the cycle Cn and it is denoted by Sn.
Definition 2.6:
A set of vertices of G is a dominating set of G if every vertices of G is adjacent to atleast one vertex of S. The dominating set of cardinality γ(G) is referred to as a minimum dominating set.
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Definition 2.7:
The minimum cardinality among the dominating sets in G is called domination number of G and is denoted by γ(G).
Definition 2.8:
The dominating set S of G is Split dominating set of G if induced sub graph
〈V − S〉 is disconnected. The split domination number γs(G) of G is referred to as a minimum cardinality of split dominating set.
Definition 2.9:
The dominating set S of G is a non-split dominating set of G if induced sub
graph 〈V − S〉 is connected. The non-split domination number γns(G) of G is referred to as a minimum cardinality of non-split dominating set.
Remarks:
a. Number of vertices in Sn is P = 2n. b. Number of Edges in Sn is q = 2n c. Maximum degree in Sn is Δ = 3. d. Minimum degree in Sn is δ = 1. e. Maximum degree in the line graph of Sn is Δ = 4. f. Minimum degree in the line graph of Sn is δ = 2
III. DOMINATION NUMBER OF LINE GRAPH OF N-SUNLET GRAPH
Proposition 3.1:
The domination number of line graph of 3-sunlet graph, γ(L(S3)) = 2.
Proof:
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Fig 1: 3-sunlet graph Fig 2: Line graph of 3-sunlet graph
Let D = {v1, v4}. The vertex v1dominates the vertices v2 and v6. The vertex v4 dominates the other vertices v3 and v5. Therefore D is the dominating set of the line graph of 3 Sunlet graph. If we remove the vertex v1 from D, v1 is not dominated by v4 and hence D can’t be a dominating set of the graph. If we remove the vertex v4 from D, v1 does not dominates the vertices v3, v4 and v5 and hence D can’t be a dominating set of the graph. Therefore D is the minimal dominating set.
The domination number line graph of 3 sunlet graph is 2.i. e. , γ(L(S3)) = 2.
Proposition 3.2:
The domination number of line graph of 4-sunlet graph, γ(L(S4)) = 2.
Proof:
Fig 3: 4-sunlet graph Fig 4: Line graph of 4-sunlet graph
Let D = {v2, v6}. The vertex v2dominates the vertices v1 and v3. The vertex v6 dominates the other vertices v5, v7 and v8. Therefore D is the dominating set of the line graph of 4 Sunlet graph. If we remove the vertex v2 from D, the neighborhood vertices v1and v3 are not dominated by vertex v6 and hence D can’t be a dominating set of the graph. If we remove the vertex v6 from D, the neighborhood vertices v5 and v7 are not dominated by vertex v2 and hence D can’t be a dominating set of the graph. Therefore D is the minimal dominating set.
The domination number line graph of 4 sunlet graph is 2.i. e. , γ(L(S4)) = 2.
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Theorem 3.3:
The domination number of line graph of n- sunlet graph, n γ(L(S )) = ⌈ ⌉. n 4
Proof:
The number of vertices of n-sunlet graph is n. Then it has 2n edges. The number of vertices of line graph of the sunlet graph is 2n. Let the vertices of the graph is labeled as {1,2,…n,n+1,n+2,…2n}. Let D = {2n,2n – 4, 2n-8,….1}. Since degree of each vertex is 4, it dominates atleast four vertices. The vertex 2n is adjacent to the vertices 1,2, 2n-1,2n-2. So these vertices are dominated by the vertex 2n. Similarly all the vertices of G is dominated by atleast one vertex in D. If any vertex 푣 ∈ 퐷 is removed from v, D cannot be a dominating set. Therefore D is the minimal dominating set. Therefore the domination number of the line graph of n- 푛 sunlet graph is ⌈ ⌉. 4
Proposition 3.4:
The Split domination number of line graph of a 3 sun-let graph, γs(L(S3)) = 2.
Proof:
Let D = {v2, v4}. By proposition 3.1, we have D to be the dominating set of line graph of S3 as all the vertices of S3 is adjacent to any one vertices of D. If ′ ′ D = D − {v2}, then on removing any one vertex from D, D can't be the dominating set of the graph. Therefore D is the minimal dominating set. Let us consider ′ ′ S3 = S3 − D. On removing the vertices of D from S3, the graph S3 is disconnected as ′ the vertex v3 becomes isolated. The dominating set S3 is Split dominating set of G as induced sub graph 〈V − D〉 is disconnected. ∴ The split domination number of 3
sun-let graph is 2.i. e. , γs(L(S3)) = 2.
Theorem 3.5:
The Split domination number of the line graph of n- sunlet graph, n γ (L(S )) = ⌈ ⌉ s n 4
Proof:
Let the vertices of the line graph of sunlet graph is labeled as {1,2,…n,n+1,n+2,…2n} . Let D = {2n, 2n – 4, …,1}Let D be dominating set of the line
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graph of sunlet graph. Let 푣 ∈ < 푉 − 퐷 >. If we cut the vertex 푣 from the graph, the graph is partitioned into blocks, which makes the graph disconnected. Therefore the dominating set D is split dominating set of G. Therefore the split domination n number of the line graph of n- sunlet graph, γ (L(S )) = ⌈ ⌉. s n 4
IV Conclusion In this paper the domination number and split domination number of line graph of n-sunlet graph have been identified.
References
[1] Cockayne, E.J., and Hedetniemi, S.T., Towards a theory of domination in graphs, Networks, Fall, 1977, 247-271.
[2] Haynes, T.W., Hedetniemi, S.T. and Slater, P.J.,Fundamentals of domination in graphs ; Marcel Dekkar, Inc-New York, 1998.
[3] Harary, F., Graph Theory, Addison – Wesley,Massachusetts, 1969.
[4] Kulli, V.R. and Janakiram, B., The split domination number of a graph; Graph theory notes of New York, XXXII, 16-19 (1997); New York Acadamy of Sciences.
[5] Laskar, R.C. and Walikar, H.B., On domination related concepts in graph theory, in : Lecture notes in Match., 885, 1981, 308-320.
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