Vertices belonging to all or to no minimum locating dominating sets of trees

Abstract

A set $D$ of vertices in a graph $G$ is a locating-dominating set if for every two vertices $u$, $v$ of $G \setminus D$ the sets $N(u) \cap D$ and $N(v) \cap D$ are non-empty and different. In this paper, we characterize vertices that are in all or in no minimum locating dominating sets in trees. The characterization guarantees that the $\gamma_L$-excellent tree can be recognized in a polynomial time.