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Eternal m-security subdivision numbers in graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 3، دوره 4، شماره 1، شهریور 2019، صفحه 25-33 اصل مقاله (411.8 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2018.25948.1058 | ||
| نویسنده | ||
| Maryam Atapour* | ||
| Department of Mathematics Faculty of basic sciences University of Bonab Bonab, Iran, Po. Box: 5551761167 | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| eternal $m$-secure set؛ eternal -security number؛ eternal m-security subdivision number | ||
| مراجع | ||
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