Locating domination in bipartite graphs and their complements

TL;DR

This paper investigates the relationship between the location-domination number of bipartite graphs and their complements, providing a characterization of graphs where this number differs by exactly one.

Contribution

It introduces a characterization of connected bipartite graphs with a specific difference in their location-domination numbers, using a novel edge-labeled graph construction.

Findings

Characterization of bipartite graphs with λ(Ḡ)=λ(G)+1

Introduction of an edge-labeled graph G^S for analysis

Insights into the structure of locating-dominating sets

Abstract

A set $S$ of vertices of a graph $G$ is \emph{distinguishing} if the sets of neighbors in $S$ for every pair of vertices not in $S$ are distinct. A \emph{locating-dominating set} of $G$ is a dominating distinguishing set. The \emph{location-domination number} of $G$, $\lambda(G)$, is the minimum cardinality of a locating-dominating set. In this work we study relationships between $\lambda({G})$ and $\lambda (\overline{G})$ for bipartite graphs. The main result is the characterization of all connected bipartite graphs $G$ satisfying $\lambda (\overline{G})=\lambda({G})+1$. To this aim, we define an edge-labeled graph $G^S$ associated with a distinguishing set $S$ that turns out to be very helpful.