TY - JOUR
ID - 13803
TI - Eternal m-security subdivision numbers in graphs
JO - Communications in Combinatorics and Optimization
JA - CCO
LA - en
SN - 2538-2128
AU - Atapour, Maryam
AD - Department of Mathematics
Faculty of basic sciences
University of Bonab
Bonab, Iran, Po. Box: 5551761167
Y1 - 2019
PY - 2019
VL - 4
IS - 1
SP - 25
EP - 33
KW - eternal $m$-secure set
KW - eternal -security number
KW - eternal m-security subdivision number
DO - 10.22049/cco.2018.25948.1058
N2 - An eternal $m$-secure set of a graph $G = (V,E)$ is a set $S_0subseteq V$ 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 $uvin E(G)$ 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.
UR - http://comb-opt.azaruniv.ac.ir/article_13803.html
L1 - http://comb-opt.azaruniv.ac.ir/article_13803_3fd4e7d1ecc4a8bed4b5eb43305015eb.pdf
ER -