Computing the metric dimension by decomposing graphs into extended biconnected components

Duygu Vietz; Stefan Hoffmann; Egon Wanke

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arXiv · 1806.10389

Computing the metric dimension by decomposing graphs into extended biconnected components

Duygu Vietz·Stefan Hoffmann·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 distances between $u$ and $w$ and the distance between $v$ and $w$ are different. The $\textit{Metric Dimension}$ of $G$ is the size of a smallest resolving set for $G$. Deciding whether a given graph $G$ has Metric Dimension at most $k$ for some integer $k$ is well-known to be NP-complete. Many research has been done to understand the complexity of this problem on restricted graph classes. In this paper, we decompose a graph into its so called $\textit{extended biconnected components}$ and present an efficient algorithm for computing the metric dimension for a class of graphs having a minimum resolving set with a bounded number of vertices in every extended biconnected component. Further we show that the decision problem METRIC DIMENSION remains NP-complete when the above limitation is extended to usual biconnected components.

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BibTeX RIS

Duygu Vietz, Stefan Hoffmann, Egon Wanke. 2018-06-27. Computing the metric dimension by decomposing graphs into extended biconnected components. https://arxiv.org/abs/1806.10389

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