Double domination critical and stable graphs upon vertex removal
Mustapha Chellali, Soufiane Khelifi · Discussiones Mathematicae Graph Theory · 2012
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 of G.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.