A Note on the Total Domination Vertex Critical Graphs

Abstract

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 exist a \(3\)-\(\gamma_t\)-critical graph \(G\) of order \(\Delta(G) + 3\) with \(\Delta(G)\) odd and \(\delta(G) \geq 2\)? In this paper, we positively answer that for each odd integer \(n \geq 9\), there exists a \(3\)-\(\gamma_t\)-critical graph \(G_n\) of order \(n+3\) with \(\delta(G) \geq 2\). On the 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 \(\delta(G) \geq 2\).