Total domination changing and stable graphs upon vertex removal

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination vertex removal stable if the removal of an arbitrary vertex leaves the total domination number unchanged. On the other hand, a graph is total domination vertex removal changing if the removal of an arbitrary vertex changes the total domination number. In this paper, we study total domination vertex removal changing and stable graphs.

► The effects of vertex removal on the total domination number γtγt of a graph are studied.
► Total domination changing graphs are characterized.
► Sharp bounds on γtγt for total domination changing graphs are determined.
► Total domination changing graphs attaining the bounds on γtγt are characterized.
► Total domination stable graphs are characterized.