Rainbow domination and related problems on some classes of perfect graphs

TL;DR

This paper studies the rainbow domination number in perfect graphs, defining a new graph labeling concept and analyzing its properties and bounds within these graph classes.

Contribution

It introduces the rainbow domination problem for perfect graphs and explores its properties, providing bounds and characterizations for these graph classes.

Findings

Defined the rainbow domination number for perfect graphs.

Established bounds and properties of rainbow domination in these graphs.

Analyzed specific classes of perfect graphs for rainbow domination.

Abstract

Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G) \rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\emptyset$, $f(N(x)) =[k]$. 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.