Locating-dominating partitions for some classes of graphs

Abstract

A dominating set of a graph $G$ is a set $D\subseteq V(G)$ such that every vertex in $V(G)\setminus D$ is adjacent to at least one vertex in $D$. A set $L\subseteq V(G)$ is a locating set of $G$ if every vertex in $V(G)\setminus L$ has pairwise distinct open neighborhoods in $L$. A set $D\subseteq V(G)$ is a locating-dominating set of $G$ if $D$ is a dominating set and a locating set of $G$. The location-domination number of $G$, denoted by ${\hat{I}}_{LD}^3(G)$, is the minimum cardinality among all locating-dominating sets of $G$. A well-known conjecture in the study of locating-dominating sets is that if $G$ is an isolate-free and twin-free graph of order $n$, then ${\hat{I}}_{LD}^3(G)\le \frac{n}{2}$. Recently, Bousquet et al. [Discrete Math. 348 (2025), 114297] proved that if $G$ is an isolate-free and twin-free graph of order $n$, then ${\hat{I}}_{LD}^3(G)\le \lceil \frac{5n}{8}\rceil$ and posed the question whether the vertex set of such a graph can be partitioned into two locating sets. We answer this question affirmatively for twin-free distance-hereditary graphs, maximal outerplanar graphs, split graphs, and co-bipartite graphs. In fact, we prove a stronger result that for any graph $G$ without isolated vertices and twin vertices, if $G$ is a distance-hereditary graph or a maximal outerplanar graph or a split graph or a co-bipartite graph, then the vertex set of $G$ can be partitioned into two locating-dominating sets. Consequently, this also confirms the original conjecture for these graph classes.