Domination and fractional domination in digraphs

TL;DR

This paper explores bounds on domination and fractional domination numbers in digraphs, establishing new relationships with the independence number and providing sharp bounds for specific classes.

Contribution

It proves that the fractional domination number is at most twice the independence number and offers a factorial bound for triangle-free digraphs, advancing understanding of domination parameters.

Findings

Fractional domination number ≤ 2 × independence number

Domination number ≤ α(G)·α(G)! for triangle-free digraphs

First bound proven to be sharp

Abstract

In this paper, we investigate the relation between the (fractional) domination number of a digraph $G$ and the independence number of its underlying graph, denoted by $\alpha(G)$. More precisely, we prove that every digraph $G$ has fractional domination number at most $2\alpha(G)$, and every directed triangle-free digraph $G$ has domination number at most $\alpha(G)\cdot \alpha(G)!$. The first bound is sharp.