Non-isolating 2-bondage in graphs

Abstract

A 2-dominating set of a graph $G = (V,E)$ is a set $D$ of vertices of $G$ such that every vertex of $V(G) \setminus D$ has at least two neighbors in $D$. The 2-domination number of a graph $G$, denoted by $\gamma_2(G)$, is the minimum cardinality of a 2-dominating set of $G$. The non-isolating 2-bondage number of $G$, denoted by $b_2'(G)$, is the minimum cardinality among all sets of edges $E' \subseteq E$ such that $\delta(G-E') \ge 1$ and $\gamma_2(G-E') > \gamma_2(G)$. If for every $E' \subseteq E$, either $\gamma_2(G-E') = \gamma_2(G)$ or $\delta(G-E') = 0$, then we define $b_2'(G) = 0$, and we say that $G$ is a $\gamma_2$-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 $\gamma_2$-non-isolatingly strongly stable trees.