Nonsplit Domination Vertex Critical Graph

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) is the minimum cardinality of a nonsplit domination set. The purpose of this paper is to initiate the investigation of those graphs which are critical in the following sense: A graph G is called vertex domination critical if γ(G−v) γ(G) for every vertex v in G. A graph G is called vertex nonsplit critical if γns(G − v) γns(G) for every vertex v in G. Initially we test whether some particular classes of graph are γns-critical or not and then we have shown that there is no existence of 2-γns-critical graph. Then 3-γns-critical graphs are characterized.