Total edge critical graphs with leaves

Let γt(G) denote the total domination number of the graph G. The graph G is said to be total domination edge critical, or simply γt(G)-critical, if γt(G+e)<γt(G) for each e∈E(G¯). We show that any γt(G)-critical graph G with γt(G)≥5 has at most γt(G)−2 leaves, and characterize those γt-critical graphs G having exactly γt(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 γt-critical unicyclic graphs.