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 $γ_{LD}(G)$, 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 $γ_{LD}(G)\le \frac{n}{2}$. 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 $γ_{LD}(G)\le \lceil\frac{5n}{8}\rceil$ 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.

翻译：图$G$的支配集是集合$D \subseteq V(G)$，使得$V(G) \setminus D$中的每个顶点至少与$D$中的一个顶点相邻。集合$L\subseteq V(G)$是$G$的定位集，如果$V(G) \setminus L$中的每个顶点在$L$中的开邻域两两不同。集合$D\subseteq V(G)$是$G$的定位-支配集，如果$D$既是$G$的支配集又是$G$的定位集。$G$的定位-支配数，记为$γ_{LD}(G)$，是$G$的所有定位-支配集中的最小基数。关于定位-支配集的一个著名猜想是：若$G$是阶数为$n$的无孤立顶点且无孪生顶点的图，则$γ_{LD}(G)\le \frac{n}{2}$。最近，Bousquet等人[Discrete Math. 348 (2025), 114297]证明了若$G$是阶数为$n$的无孤立顶点且无孪生顶点的图，则$γ_{LD}(G)\le \lceil\frac{5n}{8}\rceil$，并提出问题：这类图的顶点集能否划分为两个定位集。我们针对无孪生顶点的距离遗传图、极大外平面图、分割图和余二部图，给出了该问题的肯定回答。实际上，我们证明了更强的结论：对于任意无孤立顶点且无孪生顶点的图$G$，若$G$是距离遗传图、极大外平面图、分割图或余二部图，则$G$的顶点集可划分为两个定位-支配集。因此，这也证实了这些图类上的原始猜想。