Unicyclic graphs with strong equality between the ۲-rainbow domination and independent ۲-rainbow domination numbers

A ۲-\emph{rainbow dominating function} (۲RDF) on a graph G=(V,E) is a‎ ‎function f from the vertex set V to the set of all subsets of‎ ‎the set \{۱,۲\} such that for any vertex v\in V with‎ ‎f(v)=\emptyset the condition \bigcup_{u\in N(v)}f(u)=\{۱,۲\}‎ ‎is fulfilled‎. ‎A ۲RDF f is independent (I۲RDF) if no two vertices‎ ‎assigned nonempty sets are adjacent‎. ‎The weight of a ۲RDF‎ ‎f is the value \omega(f)=\sum_{v\in V}|f (v)|‎. ‎The‎ ‎۲-\emph{rainbow domination number} \gamma_{r۲}(G) (respectively‎, ‎the independent ۲-rainbow domination number i_{r۲}(G)‎) ‎is the minimum weight of a ۲RDF (respectively‎, ‎I۲RDF) on G‎. ‎We‎ say that \gamma_{r۲}(G) is strongly equal to i_{r۲}(G) and‎ ‎denote by \gamma_{r۲}(G)\equiv i_{r۲}(G)‎, ‎if every ۲RDF on G‎ ‎of minimum weight is an I۲RDF‎. ‎In this paper we‎ ‎characterize all unicyclic graphs G with \gamma_{r۲}(G)\equiv i_{r۲}(G)‎.

Keywords:

۲-rainbow domination number , independent ۲-rainbow domination number , strong equality , tree , unicyclic graph