A Note on Total Domination Critical Graphs

The total domination number of G denoted by γt(G) is the minimum cardinality of a total dominating set of G. A graph G is total domination vertex critical or just γt-critical, if for any vertex v of G that is not adjacent to a vertex of degree one, γt(G − v) < γt(G). If G is γt-critical and γt(G) = k, then G is k-γt-critical. Haynes et al [The diameter of total domination vertex critical graphs. Discrete Math., 286 (2004) 255-261.] proposed an open problem: Does there exist a 4-γt-critical graph with diameter 2? Afterwards Mojdeh and Rad [On an open problem concerning total domination critical graphs, Expositions Mathmatics, 25 (2007) 175-179.] posed a conjecture: For r ≥ 6, there is no 3-γt-critical r-regular graph of order 2r + 1. In this paper, we solve both problems by showing that there exists a family of 4-γt-critical graphs with diameter 2 and a family of 3-γt-critical r-regular graphs with order 2r+1 for even r.