Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity

TL;DR

This paper investigates the complexity of identifying codes, locating-dominating sets, and metric dimension in interval and permutation graphs, proving NP-completeness for decision problems and fixed-parameter tractability for metric dimension in interval graphs.

Contribution

It establishes NP-completeness for these problems on specific graph classes and shows fixed-parameter tractability of metric dimension for interval graphs.

Findings

NP-completeness for decision problems on interval and permutation graphs.

Fixed-parameter tractability of metric dimension on interval graphs.

Demonstration of computational complexity distinctions among the problems.

Abstract

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted IDENTIFYING CODE, (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.