Total domination critical and stable graphs upon edge removal Original Research Article

A set image of vertices in a graph image is a total dominating set of image if every vertex of image is adjacent to some vertex in image. The minimum cardinality of a total dominating set of image is the total domination number of image. 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.