Metric Dimension of Bounded Tree-length Graphs

TL;DR

This paper investigates the computational complexity of finding the metric dimension in graphs with bounded tree-length, extending fixed-parameter tractability results to broader graph classes like chordal and permutation graphs.

Contribution

It extends the fixed-parameter tractability of computing the metric dimension to all graphs of bounded tree-length, including several well-known graph classes.

Findings

FPT algorithm for graphs of bounded tree-length

Extension of FPT results to chordal, AT-free, and permutation graphs

FPT algorithm parameterized by modular-width

Abstract

The notion of resolving sets in a graph was introduced by Slater (1975) and Harary and Melter (1976) as a way of uniquely identifying every vertex in a graph. A set of vertices in a graph is a resolving set if for any pair of vertices x and y there is a vertex in the set which has distinct distances to x and y. A smallest resolving set in a graph is called a metric basis and its size, the metric dimension of the graph. The problem of computing the metric dimension of a graph is a well-known NP-hard problem and while it was known to be polynomial time solvable on trees, it is only recently that efforts have been made to understand its computational complexity on various restricted graph classes. In recent work, Foucaud et al. (2015) showed that this problem is NP-complete even on interval graphs. They complemented this result by also showing that it is fixed-parameter tractable (FPT) parameterized by the metric dimension of the graph. In this work, we show that this FPT result can in fact be extended to all graphs of bounded tree-length. This includes well-known classes like chordal graphs, AT-free graphs and permutation graphs. We also show that this problem is FPT parameterized by the modular-width of the input graph.