Feedback vertex sets of planar digraphs with fixed digirth
Simon Dreyer, Alexandre Pinlou, Petru Valicov

TL;DR
This paper investigates bounds on the size of minimum feedback vertex sets in planar digraphs with fixed digirth, providing new theoretical bounds, constructions, and a planar analogue of the Lucchesi-Younger theorem.
Contribution
It improves existing bounds on feedback vertex sets in planar digraphs, introduces a new construction method, and establishes a planar digraph analogue of a key cycle packing theorem.
Findings
Improved upper bound: fvs_g(n) ≤ (n-2)/(g-2) for all g ≥ 3.
Constructed infinite families of planar digraphs with large feedback vertex sets.
Narrowed the gap between lower and upper bounds for the maximum ratio of feedback vertex set size to number of vertices.
Abstract
Let denote the size of a minimum feedback vertex set of a digraph . We study , which is the maximum over all -vertex planar digraphs of digirth . It is known in the literature that and , , and for . In particular for , . We improve all lower and upper bounds starting with digirth 4. Namely, we show that for all , 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…
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