Complexity of Metric Dimension on Planar Graphs

TL;DR

This paper investigates the computational complexity of the metric dimension problem on planar graphs, establishing NP-completeness for certain classes and providing efficient algorithms for others, advancing understanding of its computational boundaries.

Contribution

It proves NP-completeness of the metric dimension problem on planar graphs with degree 6 and offers a polynomial-time solution for outerplanar graphs, clarifying complexity boundaries.

Findings

NP-complete for planar graphs of degree 6

Polynomial-time algorithm for outerplanar graphs

Clarifies complexity boundaries of the problem

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.