A dominating set of a graph G=(V,E)\documentclass[12pt]{minimal}
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\begin{document}$$G = (V,E)$$\end{document} is a set D of vertices of G such that every vertex of V(G)\D\documentclass[12pt]{minimal}
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\begin{document}$$V(G){\setminus }D$$\end{document} has a neighbor in D. The domination number of a graph G, denoted by γ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (G)$$\end{document}, is the minimum cardinality of a dominating set of G. The non-isolating bondage number of G, denoted by b′(G)\documentclass[12pt]{minimal}
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\begin{document}$$b'(G)$$\end{document}, is the minimum cardinality among all sets of edges E′⊆E\documentclass[12pt]{minimal}
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\begin{document}$$E' \subseteq E$$\end{document} such that δ(G-E′)≥1\documentclass[12pt]{minimal}
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\begin{document}$$\delta (G-E') \ge 1$$\end{document} and γ(G-E′)>γ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (G-E') > \gamma (G)$$\end{document}. If for every E′⊆E\documentclass[12pt]{minimal}
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\begin{document}$$E' \subseteq E$$\end{document} we have γ(G-E′)=γ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma (G-E') = \gamma (G)$$\end{document} or δ(G-E′)=0\documentclass[12pt]{minimal}
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\begin{document}$$\delta (G-E') = 0$$\end{document}, then we define b′(G)=0\documentclass[12pt]{minimal}
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\begin{document}$$b'(G) = 0$$\end{document}, and we say that G is a γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document}-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 γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document}-non-isolatingly strongly stable trees.
