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.