Total domination dot-critical graphs

A graph GG with no isolated vertex is total domination vertex-critical if for any vertex vv of GG that is not adjacent to a vertex of degree one, the total domination number of G−vG−v is less than the total domination number of GG. A graph is total domination dot-critical if contracting any edge decreases the total domination number. In this paper, we study total domination dot-critical graphs. We present several properties of these graphs. We show that the total domination dot-critical graphs include the total domination vertex-critical graphs.