Locating-total domination critical graphs

A locating-total dominating set of a graph G = (V (G), E(G)) with no isolated vertex is a set S ⊆ V (G) such that every vertex of V (G) is adjacent to a vertex of S and for every pair of distinct vertices u and v in V (G) − S, N(u) ∩ S = N(v) ∩ S. Let γ t (G) be the minimum cardinality of a locating-total dominating set of G. A graph G is said to be locating-total domination vertex critical if for every vertex w that is not a support vertex, γ t (G−w) < γ t (G). Locating-total domination edge critical graphs are defined similarly. In this paper, we study locating-total domination critical graphs.