Complexity of Metric Dimension on Planar Graphs
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 NPcomplete. Then, we give a polynomialtime algorithm for determining the metric dimension of outerplanar graphs.
 Publication:

arXiv eprints
 Pub Date:
 July 2011
 arXiv:
 arXiv:1107.2256
 Bibcode:
 2011arXiv1107.2256D
 Keywords:

 Computer Science  Computational Complexity;
 G.2.2
 EPrint:
 v5: minor modifications. to appear in JCSS