Independent Locating-Dominating Sets in Pseudotrees
Abstract
An ILD-set in a connected graph is a subset 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 . 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 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.
Cite
@article{arxiv.2605.07361,
title = {Independent Locating-Dominating Sets in Pseudotrees},
author = {José Cáceres and Ignacio M. Pelayo},
journal= {arXiv preprint arXiv:2605.07361},
year = {2026}
}