Total edge critical graphs with leaves

Let γ t ( G ) denote the total domination number of the graph G . The graph G is said to be total domination edge critical, or simply γ t ( G ) -critical, if γ t ( G + e ) < γ t ( G ) for each e ∈ E ( G ¯ ) . We show that any γ t ( G ) -critical graph G with γ t ( G ) ≥ 5 has at most γ t ( G ) − 2 leaves, and characterize those γ t -critical graphs G having exactly γ t ( G ) − 2 leaves. We also constructively establish the existence (with one exception) of h -critical graphs G with k leaves, where k is any nonnegative integer at most h − 2 . Finally, we characterize the γ t -critical unicyclic graphs.