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, ${\displaystyle N_G\left(u\right)\cap D\ne N_G\left(v\right)\cap D}$. The locating-total domination number ${\displaystyle {\gamma }_L^t\left(G\right)}$ 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 ${\displaystyle {\gamma }_L^{t+}}$-ER-critical graph, if ${\displaystyle {\gamma }_L^t(G-e)>{\gamma }_L^t\left(G\right)}$ for all e non-pendant edge of E. The purpose of this paper is to characterize the class of ${\displaystyle {\gamma }_L^{t+}}$-ER-critical graphs.