Eternal m- Security Subdivision Numbers in Graphs

Let be a simple graph with vertex set and edges set . A set is a dominating

set if every vertex in is adjacent to at least one vertex in . An eternal 1-secure set of a graph G is defined as a dominating set such that for any positive integer k and any sequence of vertices, there exists a sequence of guards with and either or and is a dominating set. If we take a guard on every vertex in an eternal 1-secure set, then for any sequence of attacks to vertices of the graph only by moving one guard during one of the edges adjacent with the vertex, the result set still remains secure. Now let for every sequence of attacks to vertices, all guards could move during one of the edges adjacent with the vertex and the result set still remains secure. This set is called eternal - secure set. The eternal - security number is defined as the minimum number of an eternal - secure set. secure set in G. An edge is subdivided if the edge is deleted and a new vertex is added, along with two new edges and . The eternal - security subdivision number of a graph is the minimum cardinality of a set of edges that must be subdivided (where each edge in can be subdivided at most once) in order to increase the eternal - security number of to increase the eternal m- security number of G. In this paper, we show that the eternal - security subdivision number is at most 3 for any nontrivial graph .