Bounding the rainbow domination number of a tree in terms of its annihilation number

Document Type: Research Paper

Authors

1 Azarbaijan Shahid Madani University

2 University Of West Georgia

Abstract

A $2$-rainbow dominating function (2RDF) 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 $v$‎. ‎The ‎weight of a 2RDF $f$ is the value $\omega(f)=\sum_{v\in V}|f‎ ‎(v)|$‎. ‎The $2$-rainbow  domination number of a graph $G$‎, ‎denoted by $\gamma_{r2}(G)$‎, ‎is the minimum weight of a 2RDF of G‎.
‎The annihilation number $a(G)$ is the largest integer $k$ such‎ ‎that the sum of the first $k$ terms of the non-decreasing degree‎ ‎sequence of $G$ is at most the number of edges in $G$‎. ‎In this‎ ‎paper‎, ‎we prove that for any tree $T$ with at least two vertices‎, ‎$\gamma_{r2}(T)\le a(T)+1$‎.

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