Eternal m-security bondage numbers in graphs

An eternal m-secure set of a graph G = (V,E) is a set S0 ⊆ 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 σm(G) is the minimum cardinality of an eternal m-secure set in G. The eternal m-security bondage number bσm (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.