Feedback vertex sets of planar digraphs with fixed digirth

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.