On the Complexity of Metric Dimension

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Abstract

The metric dimension of a graph G is the size of a smallest subset L ⊆ V(G) such that for any x,y ∈ V(G) there is a z ∈ 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, the computational complexity of determining the metric dimension of a graph is still very unclear. Essentially, we only know the problem to be NP-hard for general graphs, to be polynomial-time solvable on trees, and to have a logn-approximation algorithm for general graphs. 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 bounded-degree planar graphs is NP-complete. Then, we give a polynomial-time algorithm for determining the metric dimension of outerplanar graphs.

J. Díaz and M. Serna are partially supported by TIN-2007-66523 (FORMALISM) and SGR 2009-2015 (ALBCOM). O. Pottonen was supported by the Finnish Cultural Foundation. E.J. van Leeuwen is supported by ERC StG project PAAl no. 259515.