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arXiv · 1904.04243

The Fault-Tolerant Metric Dimension of Cographs

Duygu Vietz·Egon Wanke
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Abstract

A vertex set $U \subseteq V$ of an undirected graph $G=(V,E)$ is a \textit{resolving set} for $G$ if for every two distinct vertices $u,v \in V$ there is a vertex $w \in U$ such that the distance between $u$ and $w$ and the distance between $v$ and $w$ are different. A resolving set $U$ is {\em fault-tolerant} if for every vertex $u\in U$ set $U\setminus \{u\}$ is still a resolving set. {The \em (fault-tolerant) Metric Dimension} of $G$ is the size of a smallest (fault-tolerant) resolving set for $G$. The {\em weighted (fault-tolerant) Metric Dimension} for a given cost function $c: V \longrightarrow \mathbb{R}_+$ is the minimum weight of all (fault-tolerant) resolving sets. Deciding whether a given graph $G$ has (fault-tolerant) Metric Dimension at most $k$ for some integer $k$ is known to be NP-complete. The weighted fault-tolerant Metric Dimension problem has not been studied extensively so far. In this paper we show that the weighted fault-tolerant metric dimension problem can be solved in linear time on cographs.

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Duygu Vietz, Egon Wanke. 2019-04-07. The Fault-Tolerant Metric Dimension of Cographs. https://arxiv.org/abs/1904.04243

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