Geodesic
Total domination edge critical graphs with maximum diameter
Discussiones Mathematicae. Graph Theory, Tome 21 (2001) no. 2, pp. 187-205 
van der Merwe, Lucas; Mynhardt, Cristine; Haynes, Teresa. Total domination edge critical graphs with maximum diameter. Discussiones Mathematicae. Graph Theory, Tome 21 (2001) no. 2, pp. 187-205. http://geodesic.mathdoc.fr/item/DMGT_2001_21_2_a4/
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Denote the total domination number of a graph G by γₜ(G). A graph G is said to be total domination edge critical, or simply γₜ-critical, if γₜ(G+e) γₜ(G) for each edge e ∈ E(G̅). For 3ₜ-critical graphs G, that is, γₜ-critical graphs with γₜ(G) = 3, the diameter of G is either 2 or 3. We characterise the 3ₜ-critical graphs G with diam G = 3.

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