Rainbow domination and related problems on some classes of perfect graphs

Abstract

Let $k\in \mathbb{N}$ and let $G$ be a graph. A function $f:V(G)\to 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\varnothing$, $f(N(x))=[k]$. The rainbow domination number ${\gamma }_{kr}(G)$ is the minimum of ${\sum }_{x\in V(G)}|f(x)|$ over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs.