Non-isolating 2-bondage in graphs

A 2-dominating set of a graph G = (V,E) is a set D of vertices of G such that every vertex of V (G) \D has at least two neighbors in D. The 2-domination number of a graph G, denoted by γ2(G), is the minimum cardinality of a 2-dominating set of G. The non-isolating 2-bondage number of G, denoted by b2(G), is the minimum cardinality among all sets of edges E′ ⊆ E such that δ(G − E′) ≥ 1 and γ2(G − E′) > γ2(G). If for every E′ ⊆ E, either γ2(G−E) = γ2(G) or δ(G−E′) = 0, then we define b2(G) = 0, and we say that G is a γ2-non-isolatingly strongly stable graph. First we discuss the basic properties of non-isolating 2-bondage in graphs. We find the non-isolating 2-bondage numbers for several classes of graphs. Next we show that for every non-negative integer there exists a tree having such non-isolating 2-bondage number. Finally, we characterize all γ2-nonisolatingly strongly stable trees.