On the minimal feedback arc set of m-free Digraphs

TL;DR

This paper establishes an upper bound on the minimal feedback arc set size in m-free digraphs, relating it to the number of nonadjacent vertex pairs, thus generalizing previous results in the field.

Contribution

It proves a new inequality linking feedback arc set size and nonadjacent pairs in m-free digraphs, extending known bounds to a broader class of graphs.

Findings

Proves $eta(G) \\leq rac{1}{m-2} \\gamma(G)$ for m-free digraphs

Generalizes previous bounds on feedback arc sets

Provides a new relation between cycle elimination and nonadjacent pairs

Abstract

For a simple digraph $G$, let $\beta(G)$ be the size of the smallest subset $X\subseteq E(G)$ such that $G-X$ has no directed cycles, and let $\gamma(G)$ be the number of unordered pairs of nonadjacent vertices in $G$. A digraph $G$ is called $m$-free if $G$ has no directed cycles of length at most $m$. This paper proves that $\beta(G)\leq \frac{1}{m-2}\gamma(G)$ for any $m$-free digraph $G$, which generalized some known results.