@article {
author = {Atapour, Maryam},
title = {Eternal m-security subdivision numbers in graphs},
journal = {Communications in Combinatorics and Optimization},
volume = {4},
number = {1},
pages = {25-33},
year = {2019},
publisher = {Azarbaijan Shahid Madani University},
issn = {2538-2128},
eissn = {2538-2136},
doi = {10.22049/cco.2018.25948.1058},
abstract = {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 $G$. A suitable placement of the guards is called an eternal $m$-secure set. The eternal $m$-security number $\sigma_m(G)$ is the minimum cardinality among all eternal $m$-secure sets in $G$. An edge $uv\in E(G)$ 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 ${\rm sd}_{\sigma_m}(G)$ 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. },
keywords = {eternal $m$-secure set,eternal -security number,eternal m-security subdivision number},
url = {http://comb-opt.azaruniv.ac.ir/article_13803.html},
eprint = {http://comb-opt.azaruniv.ac.ir/article_13803_3fd4e7d1ecc4a8bed4b5eb43305015eb.pdf}
}