Total restrained reinforcement in graphs

In this paper we initiate the study of total restrained reinforcement in graphs. The total restrained reinforcement number in a graph GG with no isolated vertex, is the minimum number of edges that have to be added to GG so that the resulting graph has total restrained domination number less than total restrained domination number of GG. We obtain sharp bounds, exact values and characterization for the total restrained reinforcement number of a graph.