Total Roman \(\{2\}\)-reinforcement of graphs (Q2035667)

Summary: A total Roman \(\{2\}\)-dominating function (TR2DF) on a graph \(\Gamma=(V,E)\) is a function \(l:V\longrightarrow\{0,1,2\}\), satisfying the conditions that (i) for every vertex \(y\in V\) with \(l(y)=0\), either \(y\) is adjacent to a vertex labeled 2 under \(l\), or \(y\) is adjacent to at least two vertices labeled 1; (ii) the subgraph induced by the set of vertices with positive weight has no isolated vertex. The weight of a TR2DF \(l\) is the value \(\sum_{y\in V} l (y)\). The total Roman \(\{2\}\)-domination number (TR2D-number) of a graph \(\Gamma\) is the minimum weight of a TR2DF on \(\Gamma\). The total Roman \(\{2\}\)-reinforcement number (TR2R-number) of a graph is the minimum number of edges that have to be added to the graph in order to decrease the TR2D-number. In this manuscript, we study the properties of TR2R-number and we present some sharp upper bounds. In particular, we determine the exact value of TR2R-numbers of some classes of graphs.