Characterizations of Trees with Unique Minimum Locating-Dominating Sets

Abstract

A set $D$ of vertices in a graph $G=(V,E)$ is a locating-dominating set if for every two vertices $u,v$ in $V\setminus D$, the sets $N(u)\cap D$ and $N(v)\cap D$ are non-empty and different. We establish two equivalent conditions for trees with unique minimum locating-dominating sets.