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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 γ t (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 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 t+ L-ER-critical graph, if γ t L (G − e) > γ t L (G) for all e non-pendant edge of E. The purpose of this paper is to characterize the class of γ t+ L-ER-critical graphs.

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The average lower independence number i av (G) of a graph G=(V , E) is defined as 1 |V | v∈V i v (G), and the average lower domination number av (G) is defined as 1is the minimum cardinality of a maximal independent set (resp. dominating set) that contains v. We give an upper bound of i av (G) and av (G) for arbitrary graphs. Then we characterize the graphs achieving this upper bound for i av and for av respectively.

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