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

Document Type: Research Paper

Authors

1 Azarbaijan Shahid Madani University

2 University of Blida LAMDA-RO Laboratory

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 $v\in V$ 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-\emph{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 ‎characterize all unicyclic graphs $G$ with $\gamma_{r2}(G)\equiv i_{r2}(G)$‎.

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