Geodesic
Double domination critical and stable graphs upon vertex removal
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 643-657

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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.
Keywords: double domination, vertex removal critical graphs, vertex removal stable graphs 
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Khelifi, Soufiane; Chellali, Mustapha. Double domination critical and stable graphs upon vertex removal. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 643-657. http://geodesic.mathdoc.fr/item/DMGT_2012_32_4_a3/