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.