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 \(f: V(G) \rightarrow 2^{[k]}\) is a rainbow function if, for every vertex x with \(f(x)=\varnothing \), \(f(N(x)) =[k]\), where [k] denotes the integers ranging from 1 to 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.