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.