Total domination dot-stable graphs

A set SS of vertices in a graph GG is a total dominating set 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. Two vertices of GG are said to be dotted (identified) if they are combined to form one vertex whose open neighborhood is the union of their neighborhoods minus themselves. We note that dotting any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most 2. A graph is total domination dot-stable if dotting any pair of adjacent vertices leaves the total domination number unchanged. We characterize the total domination dot-stable graphs and give a sharp upper bound on their total domination number. We also characterize the graphs attaining this bound.