Independent Locating-Dominating Sets in Pseudotrees

TL;DR

This paper investigates the existence and properties of independent locating-dominating sets in various graph classes, providing new results, bounds, and algorithms, especially for trees and unicyclic graphs.

Contribution

It proves the existence of ILD-sets in trees and twin-free unicyclic graphs, and offers bounds, realization theorems, and algorithms for these graph families.

Findings

Graphs with girth at least 5 have ILD-sets.

No ILD-sets exist for some graphs with girth 4 and order ≥ 9.

Every tree and twin-free unicyclic graph contains ILD-sets.

Abstract

An ILD-set in a connected graph is a subset $S$ of vertices such that it is both independent and locating-dominating. The independent locating-dominating number of a graph G is the minimum cardinality of an ILD-set set of $G$. A well-known fact is that any graph with girth at least 5 has an ILD-set, but that is not clear for graphs with girth 3 and 4. In this work, we prove that there are graphs with no ILD-sets for any order $n\geq 9$ and girth 4, also showing some sufficient conditions for a bipartite graph to contain an ILD-set. Moreover, we focus our attention on trees and on unicyclic graphs, showing that every tree and every unicyclic graph contains ILD-sets, whenever in the latter case, it is twin-free. Finally, a number of bounds, realization theorems and algorithms to find an ILD-set in those families of graphs are provided.