Complexity of Metric Dimension on Planar Graphs

Josep Diaz; Olli Pottonen; Maria Serna; Erik Jan van Leeuwen

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arXiv · 1107.2256

Complexity of Metric Dimension on Planar Graphs

Josep Diaz·Olli Pottonen·Maria Serna·Erik Jan van Leeuwen

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Abstract

The metric dimension of a graph $G$ is the size of a smallest subset $L \subseteq V(G)$ such that for any $x,y \in V(G)$ with $x\not= y$ there is a $z \in L$ such that the graph distance between $x$ and $z$ differs from the graph distance between $y$ and $z$. Even though this notion has been part of the literature for almost 40 years, prior to our work the computational complexity of determining the metric dimension of a graph was still very unclear. In this paper, we show tight complexity boundaries for the Metric Dimension problem. We achieve this by giving two complementary results. First, we show that the Metric Dimension problem on planar graphs of maximum degree $6$ is NP-complete. Then, we give a polynomial-time algorithm for determining the metric dimension of outerplanar graphs.

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BibTeX RIS

Josep Diaz, Olli Pottonen, Maria Serna, Erik Jan van Leeuwen. 2011-07-12. Complexity of Metric Dimension on Planar Graphs. https://arxiv.org/abs/1107.2256

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