Identification, location-domination and metric dimension on interval and permutation graphs. I. Bounds

TL;DR

This paper establishes tight bounds for identifying, locating-dominating, and metric dimension problems on interval, permutation graphs, and cographs, providing both theoretical bounds and linear-time algorithms for cographs.

Contribution

It provides new tight bounds for solution set sizes on specific graph classes and introduces linear-time algorithms for cographs.

Findings

Tight lower bounds for solution sets on interval and permutation graphs.

Bounds are quadratic root or linear in the number of vertices.

Linear-time algorithms for cographs.

Abstract

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets of an interval or a permutation graph. In these problems, one asks to find a subset of vertices, normally called a \emph{solution} set, using which all vertices of the graph are distinguished. The identification can be done by considering the neighborhood within the solution set, or by employing the distances to the solution vertices. Normally the goal is to minimize the size of the solution set then. Here we study the case of interval graphs, unit interval graphs, (bipartite) permutation graphs and cographs. For these classes of graphs we give tight lower bounds for the size of such solution sets depending on the order of the input graph. While such lower bounds for the general class of graphs are in logarithmic order, the improved bounds in these special classes are of the order of either quadratic root or linear in terms of number of vertices. Moreover, the results for cographs lead to linear-time algorithms to solve the considered problems on inputs that are cographs.