On the Total Restrained Domination Edge Critical Graphs

Abstract

Let $G=(V,E)$ be a graph. A set $D\subseteq V$ is a total restrained dominating set of $G$ if every vertex in $V$ has a neighbor in $D$ and every vertex in $V–D$ has a neighbor in $V–D$. The cardinality of a minimum total restrained dominating set in $G$ is the total restrained domination number of $G$. In this paper, we define the concept of total restrained domination edge critical graphs, find a lower bound for the total restrained domination number of graphs, and constructively characterize trees having their total restrained domination numbers achieving the lower bound.