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.