Unicyclic graphs with strong equality between the 2-rainbow domination and independent 2-rainbow domination numbers

Abstract

A $2$-emph{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 $vin V$ with‎ ‎$f(v)=emptyset$ the condition $bigcup_{uin N(v)}f(u)={1,2}$‎ ‎is fulfilled‎. ‎A 2RDF $f$ is independent (I2RDF) if no two vertices‎ ‎assigned nonempty sets are adjacent‎. ‎The <em>weight</em> of a 2RDF‎ ‎$f$ is the value $omega(f)=sum_{vin V}|f (v)|$‎. ‎The‎ ‎2-emph{rainbow domination number} $gamma_{r2}(G)$ (respectively‎, ‎the <em>independent $2$-rainbow domination number</em> $i_{r2}(G)$‎) ‎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 ‎characterize all unicyclic graphs $G$ with $gamma_{r2}(G)equiv i_{r2}(G)$‎.