Characterizing extremal graphs for open neighbourhood location-domination

TL;DR

This paper characterizes graphs where the only open neighbourhood locating-dominating set is the entire vertex set, identifying them as graphs with all components being half-graphs, correcting previous literature.

Contribution

It provides a precise characterization of extremal graphs for open neighbourhood locating-domination, specifically those composed solely of half-graph components.

Findings

Graphs with only the entire vertex set as an open neighbourhood locating-dominating set are exactly those with half-graph components.

Corrects a previous incorrect characterization in the literature.

Establishes a clear structural description of extremal graphs for this property.

Abstract

An open neighbourhood locating-dominating set is a set $S$ of vertices of a graph $G$ such that each vertex of $G$ has a neighbour in $S$, and for any two vertices $u,v$ of $G$, there is at least one vertex in $S$ that is a neighbour of exactly one of $u$ and $v$. We characterize those graphs whose only open neighbourhood locating-dominating set is the whole set of vertices. More precisely, we prove that these graphs are exactly the graphs all whose connected components are half-graphs (a half-graph is a special bipartite graph with both parts of the same size, where each part can be ordered so that the open neighbourhoods of consecutive vertices differ by exactly one vertex). This corrects a wrong characterization from the literature.