Publications Repository - Gdańsk University of Technology

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.

Authors

• dr inż. Marcin Krzywkowski link open in new tab