Split Domination Vertex Critical and Edge Critical Graphs

Abstract

A dominating set D of a graph G = (V;E) is a split dominating set if the induced graph hV 􀀀 Di is disconnected. The split domination number s(G) is the minimum cardinality of a split domination set. A graph G is called vertex split domination critical if s(G􀀀v) < s(G) for every vertex v 2 G. A graph G is called edge split domination critical if s(G + e) < s(G) for every edge e in G. In this paper, whether for some standard graphs are split domination vertex critical or not are investigated and then characterized 2- ns-critical and 3- ns-critical graphs with respect to the diameter of a graph G with vertex removal. Further, it is shown that there is no existence of s-critical graph for edge addition.