A characterization of locating-domination edge critical graphs

A locating-dominating set D of a graph G = (V (G), E(G)) is a set D ⊆ V (G) such that every vertex of V (G) − D is adjacent to a vertex of D and for every pair of distinct vertices u, v in V (G) − D, N(u) ∩ D = N(v) ∩ D. The minimum cardinality of a locating-dominating set is denoted by γL(G). A graph G is said to be a locating domination edge removal critical graph, or just γ L -ER-critical graph, if γL(G−e) > γL(G) for all e ∈ E(G). The purpose of this paper is to characterize the class of γ L -ER-critical graphs.