Nonsplit Domination Edge Critical Graphs

A set of vertices S is said to dominate the graph G if for each v / ∈ S, there is a vertex u ∈ S with u adjacent to v. The minimum cardinality of any dominating set is called the domination number of the graph G and is denoted by γ(G). A dominating set D of a graph G = (V,E) is a nonsplit dominating set if the induced graph 〈V − D〉 is connected. The nonsplit domination number γns(G) of the graph G is the minimum cardinality of a nonsplit domination set. The aim of this paper is to investigate of those graphs which are critical in the sense that: A graph G is called edge domination critical if γ(G + e) < γ(G) for every edge e in G. A graph G is called edge nonsplit domination critical if γns(G + e) < γns(G) for every edge e in G. Initially we verify whether some particular classes of graphs are γns critical or not. Later 2-γns-critical and 3-γns-critical graphs are characterized.