Eternal m-security subdivision numbers in graphs

An eternal m-secure set of a graph G = (V,E) is a set S_0 C V that can defend against any sequence of single-vertex attacks by means of multiple-guard shifts along the edges of G. A suitable placement of the guards is called an eternal m- secure set. The eternal m-security number σm(G) is the minimum cardinality among all eternal m-secure sets in G. An edge uv 2 E(G) is subdivided if we delete the edge uv from G and add a new vertex x and two edges ux and vx. The eternal m- security subdivision number sdσ_m(G) of a graph G is the minimum cardinality of a set of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the eternal m-security number of G. In this paper, we study the eternal m-security subdivision number in trees. In particular, we show that the eternal m-security subdivision number of trees is at most 2 and we characterize all trees attaining this bound.