Properties of total restrained domination vertex critical graphs

A graph G with no isolated vertex is total restrained domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total restrained domination number of G v is less than the total restrained domination number of G. We call these graphs gamma tr-vertex critical. If such a graph G has total restrained domination number k, then we call it k vertex critical. In this paper, we study some properties in gamma tr-vertex critical graphs of minimum degree at least two.