Properties of Total Restrained Domination Vertex Critical Graphs

Abstract

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 $k$-${\gamma }_{tr}$-vertex critical. In this paper, we study some properties in ${\gamma }_{tr}$-vertex critical graphs of minimum degree at least two.