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.

翻译：设$fvs(G)$表示有向图$G$的最小反馈顶点集的大小。我们研究$fvs_g(n)$，即所有含$n$个顶点、有向围长为$g$的平面有向图$G$中$fvs(G)$的最大值。文献中已知：$\lfloor\frac{n-1}{g-1}\rfloor \le fvs_g(n)$，且$fvs_3(n)\le \frac{3n}{5}$、$fvs_4(n)\le \frac{n}{2}$、$fvs_5(n)\le \frac{2n-5}{4}$，对于$g \ge 6$有$\lfloor\frac{n-1}{g-1}\rfloor \le fvs_g(n) \le \frac{2n-6}{g}$。特别地，当$g \ge 6$时，$\frac{1}{g-1}\le \sup_{n \ge 1} \frac{fvs_g(n)}{n} \le \frac{2}{g}$。我们从有向围长4开始改进所有上下界。具体地，我们证明对所有$g\geq 3$有$fvs_g(n)\le \frac{n-2}{g-2}$，这是通过证明最小反馈顶点集不超过一类特殊有向环的最大包装。最后这一结果是有向平面图的著名Lucchesi-Younger定理的类比，具有独立意义。另一方面，我们开发了一种新工具，通过连接弧不相交的有向环来构造固定有向围长且$fvs$较大的平面有向图。利用该工具，我们构造了有向围长$g\ge 4$且$fvs$较大的无穷平面有向图族。这些构造结合我们的上界表明：对所有$g \ge 6$（除$g=7$其下界不同外），有$\frac{g+2}{g^2} \le \sup_{n \ge 1} \frac{fvs_g(n)}{n} \le \frac{1}{g-2}$。从而我们将$\sup_{n \ge 1} \frac{fvs_g(n)}{n}$的上下界差距从$\frac{g-2}{g(g-1)}$缩小到$\frac{4}{g^2(g-2)}$。对于$g=7$，该差距从$\frac{5}{42}$降至$\frac{1}{55}$。对于有向围长4和5，两个改进均为加法常数级别。