Small feedback vertex sets in planar digraphs

TL;DR

This paper establishes improved bounds on the size of small feedback vertex sets in directed planar graphs with girth at least 4, 5, or greater, contributing to the understanding of cycle elimination in such graphs.

Contribution

It provides tighter upper bounds on feedback vertex set sizes in directed planar graphs based on girth, improving upon previous results.

Findings

For girth 4, feedback vertex set size ≤ (5n-5)/9.

For girth 5, feedback vertex set size ≤ (2n-5)/4.

For girth ≥ 6, feedback vertex set size ≤ (2n-6)/g.

Abstract

Let $G$ be a directed planar graph on $n$ vertices, with no directed cycle of length less than $g\ge 4$. We prove that $G$ contains a set $X$ of vertices such that $G-X$ has no directed cycle, and $|X|\le \tfrac{5n-5}9$ if $g=4$, $|X|\le \tfrac{2n-5}4$ if $g=5$, and $|X|\le \tfrac{2n-6}{g}$ if $g\ge 6$. This improves recent results of Golowich and Rolnick.