Extremal Problems on Forest Cuts and Acyclic Neighborhoods in Sparse Graphs

TL;DR

This paper improves bounds on the number of edges in sparse connected graphs that guarantee the existence of a vertex cut inducing a forest, advancing understanding of acyclic neighborhoods in such graphs.

Contribution

The authors prove a tighter bound of less than 2.25n edges for the existence of a forest-inducing vertex cut in connected graphs, refining previous results.

Findings

Established that graphs with fewer than 2.25n edges have a forest-inducing vertex cut.

Improved upon previous bounds of 3n-6 edges for such cuts.

Explored weaker problem variants for potential further improvements.

Abstract

Chernyshev, Rauch, and Rautenbach proved that every connected graph on $n$ vertices with less than $\frac{11}{5}n-\frac{18}{5}$ edges has a vertex cut that induces a forest, and conjectured that the same remains true if the graph has less than $3n-6$ edges. We improve their result by proving that every connected graph on $n$ vertices with less than $\frac{9}{4}n$ edges has a vertex cut that induces a forest. We also study weaker versions of the problem that might lead to an improvement on the bound obtained.