Algorithmica

, Volume 78, Issue 3, pp 914–944

# Identification, Location-Domination and Metric Dimension on Interval and Permutation Graphs. II. Algorithms and Complexity

• Florent Foucaud
• George B. Mertzios
• Reza Naserasr
• Aline Parreau
• Petru Valicov
Article

## Abstract

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted Identifying Code, (Open) Open Locating-Dominating Set and Metric Dimension) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diameter 2 and permutation graphs of diameter 2. While Identifying Code and (Open) Locating-Dominating Set are trivially fixed-parameter-tractable when parameterized by solution size, it is known that in the same setting Metric Dimension is W[2]-hard. We show that for interval graphs, this parameterization of Metric Dimension is fixed-parameter-tractable.

## Keywords

Metric dimension Resolving set Identifying code Locating-dominating set Interval graph Permutation graph Complexity

## Notes

### Acknowledgments

We thank Adrian Kosowski for helpful preliminary discussions on the topic of this paper. We are also grateful to the reviewers for their useful comments which subsequently made the paper clearer.

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## Authors and Affiliations

• Florent Foucaud
• 1
• George B. Mertzios
• 2
• Reza Naserasr
• 3
• Aline Parreau
• 4
Email author
• Petru Valicov
• 5
1. 1.LIMOS - CNRS UMR 6158Université Blaise PascalClermont-FerrandFrance
2. 2.School of Engineering and Computing SciencesDurham UniversityDurhamUK
3. 3.CNRS - IRIF, Université Paris DiderotParisFrance
4. 4.University Lyon, Université Claude Bernard Lyon 1, CNRS, LIRIS, UMR 5205VilleurbanneFrance
5. 5.Aix-Marseille Université, CNRS, LIF, UMR 7279MarseilleFrance