Total domination critical and stable graphs upon edge removal

A set SS of vertices in a graph GG is a total dominating set of GG if every vertex of GG is adjacent to some vertex in SS. The minimum cardinality of a total dominating set of GG is the total domination number of GG. A graph is total domination edge critical if the removal of any arbitrary edge increases the total domination number. On the other hand, a graph is total domination edge stable if the removal of any arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge critical graphs. We also investigate various properties of total domination edge stable graphs.