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A dominating set of a graph \(G = (V,E)\) is a set D of vertices of G such that every vertex of \(V(G){\setminus }D\) has a neighbor in D. The domination number of a graph G, denoted by \(\gamma (G)\), is the minimum cardinality of a dominating set of G. The non-isolating bondage number of G, denoted by \(b'(G)\), is the minimum cardinality among all sets of edges \(E' \subseteq E\) such that \(\delta (G-E') \ge 1\) and \(\gamma (G-E') > \gamma (G)\). If for every \(E' \subseteq E\) we have \(\gamma (G-E') = \gamma (G)\) or \(\delta (G-E') = 0\), then we define \(b'(G) = 0\), and we say that G is a \(\gamma \)-non-isolatingly strongly stable graph. First we discuss various properties of non-isolating bondage in graphs. We find the non-isolating bondage numbers for several classes of graphs. Next we show that for every non-negative integer, there exists a tree having such non-isolating bondage number. Finally, we characterize all \(\gamma \)-non-isolatingly strongly stable trees.

Non-isolating 2-bondage in graphs