Bounding the rainbow domination number of a tree in terms of its annihilation number
نویسندگان
1 Azarbaijan Shahid Madani University
2 Azarbaijan Shahid Madani University
3 University Of West Georgia
doi
10.22108/toc.2013.3051چکیده
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, $\gamma_{r2}(T)\le a(T)+1$.
