International Symposium on Mathematical Foundations of Computer Science

MFCS 2015: Mathematical Foundations of Computer Science 2015 pp 115-126

# Metric Dimension of Bounded Width Graphs

• Rémy Belmonte
• Fedor V. Fomin
• Petr A. Golovach
• M. S. Ramanujan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9235)

## 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.

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## Authors and Affiliations

• Rémy Belmonte
• 1
• Fedor V. Fomin
• 2
• Petr A. Golovach
• 2
• M. S. Ramanujan
• 2
1. 1.Department of Architecture and Architectural EngineeringKyoto UniversityKyotoJapan
2. 2.Department of InformaticsUniversity of BergenBergenNorway