Locating-dominating sets in twin-free graphs

A locating-dominating set of a graph $G$ is a dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of $G$, denoted $\gamma_L(G)$, is the minimum cardinality of a locating-dominating set in $G$. It is conjectured [D. Garijo, A. Gonz\'alez and A. M\'arquez. The difference between the metric dimension and the determining number of a graph. Applied Mathematics and Computation 249 (2014), 487--501] that if $G$ is a twin-free graph of order $n$ without isolated vertices, then $\gamma_L(G)\le \frac{n}{2}$. We prove the general bound $\gamma_L(G)\le \frac{2n}{3}$, slightly improving over the $\lfloor\frac{2n}{3}\rfloor+1$ bound of Garijo et al. We then provide constructions of graphs reaching the $\frac{n}{2}$ bound, showing that if the conjecture is true, the family of extremal graphs is a very rich one. Moreover, we characterize the trees $G$ that are extremal for this bound. We finally prove the conjecture for split graphs and co-bipartite graphs.