On the Rainbow Restrained Domination Number

Abstract

A \(2\)-rainbow dominating function of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all subsets of the set \(\{1,2\}\) such that for any vertex \(v \in V(G)\) with \(f(v) = \emptyset\) the condition \(\bigcup_{u \in N(v)} f(u) = \{1,2\}\) is fulfilled, where \(N(v)\) is the open neighborhood of \(v\). A rainbow dominating function \(f\) is said to be a rainbow restrained domination function if the induced subgraph of \(G\) by the vertices with label \(\emptyset\) has no isolated vertex. The weight of a rainbow restrained dominating function is the value \(w(f) = \sum_{v \in V(G)} |f(v)|\). The minimum weight of a rainbow restrained dominating function of \(G\) is called the rainbow restrained domination number of \(G\). In this paper, we continue the study of the rainbow restrained domination number. First, we classify all graphs \(G\) of order \(n\) whose rainbow restrained domination number is \(n-1\). Then, we establish an upper bound on the rainbow restrained domination number of trees.