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 (Ḡ). For
3 t-critical graphs G, that is, γt-critical graphs with γt (G)= 3, the diameter of G is either 2 or 3.
We study the 3t-critical graphs G with diam G= 2.

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.