Total domination dot-critical graphs Original Research Article

A graph image with no isolated vertex is total domination vertex-critical if for any vertex image of image that is not adjacent to a vertex of degree one, the total domination number of image is less than the total domination number of image. 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.