Eternal m-Security Bondage Numbers in Graphs

An eternal m-secure set of a graph $ G = (V,E) $ is a set $ S_0 \subseteq V $ that can defend against any sequence of single-vertex attacks by means of multiple guard shifts along the edges of $ G $. The eternal m-security number $ \sigma_m (G) $ is the minimum cardinality of an eternal m-secure set in $G$. The eternal m-security bondage number $ b_{\sigma_m} (G) $ of a graph $G$ is the minimum cardinality of a set of edges of $G$ whose removal from $G$ increases the eternal m-security number of $G$. In this paper, we study properties of the eternal m-security bondage number. In particular, we present some upper bounds on the eternal m-security bondage number in terms of eternal m-security number and edge connectivity number, and we show that the eternal m-security bondage number of trees is at most 2 and we classify all trees attaining this bound.