Strong Equality Between the 2-Rainbow Domination and Independent 2-Rainbow Domination Numbers in Trees

Abstract

A 2-rainbow dominating function (2RDF) on a graph $G=(V, E)$ is a function f from the vertex set V to the set of all subsets of the set $\{1,2\}$ such that for any vertex $v\in V$ with $f(v)=\emptyset$ the condition $\bigcup _{u\in N(v)}f(u)=\{1,2\}$ is fulfilled. A 2RDF f is independent (I2RDF) if no two vertices assigned nonempty sets are adjacent. The weight of a 2RDF f is the value $\omega (f)=\sum _{v\in V}|f (v)|$. The 2-rainbow domination number $\gamma _{r2}(G)$ (respectively, the independent 2-rainbow domination number $i_{r2}(G)$) is the minimum weight of a 2RDF (respectively, I2RDF) on G. We say that $\gamma _{r2}(G)$ is strongly equal to $i_{r2}(G)$ and denote by $\gamma _{r2}(G)\equiv i_{r2}(G)$, if every 2RDF on G of minimum weight is an I2RDF. In this paper, we provide a constructive characterization of trees T with $\gamma _{r2}(T)\equiv i_{r2}(T)$.