Minimum Size of Feedback Vertex Sets of Planar Graphs of Girth at least Five

TL;DR

This paper establishes tight upper bounds on the size of minimum feedback vertex sets in planar graphs with girth at least five, advancing theoretical understanding and confirming parts of existing conjectures.

Contribution

It provides the first tight bounds for such graphs, improving previous results and confirming a specific case of a conjecture related to feedback vertex sets.

Findings

Feedback vertex set size at most (2m - n + 2)/7 for planar graphs of girth ≥ 5

Feedback vertex set size at most m/5 and (n-2)/3 derived from Euler's formula

Progress towards conjectures of Kowalik, L"uzar, and Srekovski

Abstract

A feedback vertex set of a graph is a subset of vertices intersecting all cycles. We provide tight upper bounds on the size of a minimum feedback vertex set in planar graphs of girth at least five. We prove that if $G$ is a connected planar graph of girth at least five on $n$ vertices and $m$ edges, then $G$ has a feedback vertex set of size at most $\frac{2m-n+2}{7}$. By Euler's formula, this implies that $G$ has a feedback vertex set of size at most $\frac{m}{5}$ and $\frac{n-2}{3}$. These results not only improve a result of Dross, Montassier and Pinlou and confirm the girth-5 case of one of their conjectures, but also make the best known progress towards a conjecture of Kowalik, Lu\v{z}ar and \v{S}krekovski and solves the subcubic case of their conjecture. An important step of our proof is providing an upper bound on the size of minimum feedback vertex sets of subcubic graphs with girth at least five with no induced subdivision of members of a finite family of non-planar graphs.