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