Domination stability in graphs

For a graph G=(V,E)G=(V,E), a subset D⊆V(G)D⊆V(G) is a dominating set if every vertex of V(G)∖DV(G)∖D has a neighbor in DD. The domination number of GG is the minimum cardinality of a dominating set of GG. The domination stability, or just γγ-stability, of a graph GG is the minimum number of vertices whose removal changes the domination number. We show that the γγ-stability problem is NP-hard even when restricted to bipartite graphs. We obtain several bounds, exact values and characterizations for the γγ-stability of a graph, and we characterize the trees with stγ(T)=2stγ(T)=2.