Eternal m-security subdivision numbers in graphs

Document Type : Original paper


Department of Mathematics Faculty of basic sciences University of Bonab Bonab, Iran, Po. Box: 5551761167


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.


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