DOUBLE DOMINATION CRITICAL AND STABLE GRAPHS UPON VERTEX REMOVAL

In a graph a vertex is said to dominate itself and all its neighbors. A double dominating set of a graph G is a subset of vertices that dominates every vertex of G at least twice. The double domination number of G, denoted (gamma x2)(G), is the minimum cardinality among all double dominating sets of G. We consider the effects of vertex removal on the double domination number of a graph. A graph G is (gamma x2)-vertex critical graph ((gamma x2)-vertex stable graph, respectively) if the removal of any vertex different from a support vertex decreases (does not change, respectively) (gamma x2)(G). In this paper we investigate various properties of these graphs. Moreover, we characterize (gamma x2)-vertex critical trees and (gamma x2)-vertex stable trees.