A Survey on the Effect of Graph Operations on the Domination Number of a Graph

Abstract

Abstract-A dominating set of G is a set of vertices of G such that every vertex in V -D is adjacent to a vertex in D.The domination number of G, denoted by  ( G ), is the minimum cardinality of a dominating set. Different kinds of graph operations have different outcomes on the domination number of a graph. In this paper we present a brief survey on the impact of different kinds of graph operation on the domination number of a graph.Keyword -Domination, Domination Number, Graph Operation. I. INTRODUCTIONThe study of domination number in graph theory was introduced by Claude Berge in 1958, on his book "Theory of Graphs and its Applications" [1]. But domination number was studied in the name of coefficient of external stability. At the first time the name dominating set and domination number was used by Oyestein Ore in [2]. At the beginning stage the domination number was denoted by d ( G ). Later, in 1977, the notation  ( G ) is used to denote the domination number in Since the evolution of domination theory, types of domination are also established and studied by many researchers. But the original dominating set is the base for all types of domination. Different kinds of graph operations change the graphs and hence influence the domination number of the graph also. In this survey we concentrate on the effect of various kinds of graph operations on the domination number. There are several results relating graph operations and different types of domination. But we restrict our survey only to graph operations and dominating sets. Several results are available in this regard, and many have been omitted in this brief survey. We apologize to the authors for the omission. Many results omitted here can be found in [4][5] and in the articles in the reference section. This survey is restricted to presentation of possible results that describe the effects of the graph operations on the domination number of a graph. The connectivity k = k ( G ) of a graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph. A set of independent edges in a graph G is called a matching of G. A 1 -factor or a perfect matching of a graph is a partition of its vertices into adjacent pairs. II. GRAPH THEORY TERMINOLOGY AND CONCEPTSP n , C n , K n , denotes the path, cycle and complete graph with n vertices respectively. The wheel graph is denoted by W n , and it is defined as K 1 + C n -1 . We denote the radius and diameter of G by rad( G ) and diam ( G ), respectively. The minimum and maximum degree on the vertices of G is denoted by  ( G ) and  ( G ) respectively. If  ( G ) =  ( G ) = r, then all points have the same degree and G is called regular of degree r. The regular graphs are those of degree 3, such graphs are called cubic. The complement G of a graph G also has V ( G ) as its vertex set, but two vertices are adjacent in G if and only if they are not adjacent in G.

On the \rho-subdivision number of graphs

“…Velammal in 1997 [8] and Arumugam in 2000 (see [4]) studied the domination subdivision number sd γ (G) where γ(G) is the domination number of G. This was later the topic of various papers (see e.g. Section III.H in the survey [9]). Subsequently this concept was also investigated for the stability number β(G) [4] and other invariants related to the domination number (see e.g.…”