A lower bound on the order of the largest induced forest in planar graphs with high girth

TL;DR

This paper establishes improved upper bounds on the size of minimum feedback vertex sets in planar graphs with high girth and in 2-connected graphs with maximum degree 3, advancing understanding of graph structure constraints.

Contribution

It introduces tighter bounds on feedback vertex set sizes for specific classes of graphs, improving upon previous trivial bounds.

Findings

Planar graphs with girth g have feedback vertex sets of size at most 4m/(3g).

2-connected graphs with max degree 3 have feedback vertex sets of size at most (n+2)/3.

Provides theoretical bounds that refine existing estimates.

Abstract

We give here new upper bounds on the size of a smallest feedback vertex set in planar graphs with high girth. In particular, we prove that a planar graph with girth $g$ and size $m$ has a feedback vertex set of size at most $\frac{4m}{3g}$, improving the trivial bound of $\frac{2m}{g}$. We also prove that every $2$-connected graph with maximum degree $3$ and order $n$ has a feedback vertex set of size at most $\frac{n+2}{3}$.