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 γ×2(G), is the minimum cardinality among all double dominating sets ofG. We consider the effects of vertex removal on the double domination number of a graph. A graph G is γ×2-vertex critical graph (γ×2-vertex stable graph, respectively) if the removal of any vertex different from a support vertex decreases (does not change, respectively) γ×2(G). In this paper we investigate various properties of these graphs. Moreover, we characterize γ×2-vertex critical trees and γ×2-vertex stable trees.