A note on the total domination vertex critical graphs

In 2006, Mojdeh and Jafari rad [On the total domination critical graphs, Electronic Notes in Discrete Mathematics, 24 (2006), 89-92] gave an open problem: Does there exists a 3-gamma t-critical graph G of order Delta(G) + 3 with Delta(G) odd and delta(G) >= 2 ? In this paper, we positively answer that for each odd integer n >= 9, there exists a 3-gamma(t)-critical graphs G. of order n + 3 with delta(G) >= 2. On contrary, we also prove that for Delta(G) = 3, 5, 7, there is no 3-gamma(t)-critical graph of order Delta(G) + 3 with delta(G) >= 2.