Locating-dominating sets and identifying codes in graphs of girth at least 5

TL;DR

This paper investigates the minimal size of locating-dominating sets and identifying codes in graphs with girth at least 5 and specified minimum degree, providing bounds and constructions.

Contribution

It introduces bounds on the size of these sets in graphs with girth ≥ 5 using vertex-disjoint paths and constructs near-optimal examples.

Findings

Upper bounds on set sizes established

Graphs constructed close to these bounds

Technique of vertex-disjoint paths employed

Abstract

Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meet these bounds.