Geodesic
A characterization of locating-total domination edge critical graphs
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 1, pp. 197-202

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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, N_G(u) ∩ D ≠ N_G(v) ∩ D. The locating-total domination number γ_L^t(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^t+-ER-critical graph, if γ_L^t(G-e) > γ_L^t(G) for all e non-pendant edge of E. The purpose of this paper is to characterize the class of γ_L^t+-ER-critical graphs.
Keywords: locating-domination, critical graph 
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Blidia, Mostafa; Dali, Widad. A characterization of locating-total domination edge critical graphs. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 1, pp. 197-202. http://geodesic.mathdoc.fr/item/DMGT_2011_31_1_a12/