A characterization of locating-total domination edge critical graphs

For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number γt(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V (G) \ D, NG(u)∩D 6= NG(v)∩D. The locating-total domination number γ t L (G) is the minimum cardinality of a locating-total dominating set of G. A graph G is said to be a locating-total domination edge removal critical graph, or just a γ L -ER-critical graph, if γ L (G − e) > γ L (G) for all e non-pendant edge of E. The purpose of this paper is to characterize the class of γ L -ER-critical graphs.