Forest Cuts in Sparse Graphs

TL;DR

This paper explores a conjecture that sparse graphs with fewer than 3n-6 edges contain a vertex cut inducing a forest, verifying it for planar graphs and improving bounds for general graphs.

Contribution

The paper introduces a new conjecture about vertex cuts in sparse graphs and proves it for planar graphs, providing improved bounds for general graphs.

Findings

Verified the conjecture for planar graphs.

Established a bound of (11/5)n - (18/5) edges for general graphs.

Showed maximal planar graphs lack such vertex cuts.

Abstract

We propose the conjecture that every graph $G$ of order $n$ with less than $3n-6$ edges has a vertex cut that induces a forest. Maximal planar graphs do not have such vertex cuts and show that the density condition would be best possible. We verify the conjecture for planar graphs and show that every graph $G$ of order $n$ with less than $\frac{11}{5}n-\frac{18}{5}$ edges has a vertex cut that induces a forest.