Feedback vertex sets of planar digraphs with fixed digirth

2026-05-12

AI Summary

This study investigates the maximum possible size of a minimum feedback vertex set (FVS) in planar directed graphs with fixed girth \( g \). By establishing a Lucchesi–Younger-type theorem—bounding the FVS size by the maximum number of arc-disjoint directed cycles—and introducing novel graph construction techniques to interconnect such cycles, the work bridges combinatorial optimization and cycle packing theory to significantly narrow the gap between upper and lower bounds on \( fvs_g(n)/n \). Improved bounds are achieved for all \( g \geq 4 \); notably, for \( g = 7 \), the upper bound is reduced from \( 5/42 \) to \( 1/55 \), and for \( g \geq 6 \), the gap is tightened to \( 4/(g^2(g-2)) \).

Abstract

Let $fvs(G)$ denote the size of a minimum feedback vertex set of a digraph $G$. We study $fvs_g(n)$, which is the maximum $fvs(G)$ over all $n$-vertex planar digraphs $G$ of digirth $g$. It is known in the literature that $\lfloor\frac{n-1}{g-1}\rfloor \le fvs_g(n)$ and $fvs_3(n)\le \frac{3n}{5}$, $fvs_4(n)\le \frac{n}{2}$, $fvs_5(n)\le \frac{2n-5}{4}$ and $\lfloor\frac{n-1}{g-1}\rfloor \le fvs_g(n) \le \frac{2n-6}{g}$ for $g \ge 6$. In particular for $g \ge 6$, $\frac{1}{g-1}\le \sup_{n \ge 1} \frac{fvs_g(n)}{n} \le \frac{2}{g}$. We improve all lower and upper bounds starting with digirth 4. Namely, we show that $fvs_g(n)\le \frac{n-2}{g-2}$ for all $g\geq 3$, by proving that the minimum feedback vertex set is at most the maximum packing of a special type of directed cycles. This last result is a planar-digraph analogue of the celebrated Lucchesi-Younger theorem and is of independent interest. On the other hand, we develop a new tool to construct planar digraphs of fixed digirth and large $fvs$ by connecting arc-disjoint directed cycles. Using it, we provide constructions of infinite families of planar digraphs of digirth $g\ge 4$ and large $fvs$. These constructions together with our upper bound show that $\frac{g+2}{g^2} \le \sup_{n \ge 1} \frac{fvs_g(n)}{n} \le \frac{1}{g-2}$ for all values $g \ge 6$, except $g =7$, for which the lower bound is different. We thus decrease the gap between the lower and the upper bound for $\sup_{n \ge 1} \frac{fvs_g(n)}{n}$ from $\frac{g-2}{g(g-1)}$ to $\frac{4}{g^2(g-2)}$. For $g = 7$ this gap goes from $\frac{5}{42}$ to $\frac{1}{55}$. For digirth 4 and 5, both improvements are by an additive constant.

Problem

• feedback vertex set
• planar digraph
• digirth
• extremal graph theory

Innovation

• feedback vertex set
• planar digraph
• digirth
• cycle packing
• Lucchesi-Younger theorem