On the minimal feedback arc set of m-free Digraphs

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 $\beta (G)\le \frac{1}{m-2}\gamma (G)$ for any $m$-free digraph $G$, which generalized some known results.