On forest and bipartite cuts in sparse graphs

TL;DR

This paper establishes new sufficient conditions for the existence of forest and bipartite vertex cuts in sparse graphs, improving previous bounds and contributing to the understanding of graph cut thresholds.

Contribution

It provides improved bounds for the existence of forest and bipartite cuts in sparse graphs, advancing the theoretical understanding of graph partitioning.

Findings

Graphs with fewer than (19n - 28)/8 edges have a forest cut.

Graphs with fewer than (80n - 134)/31 edges have a bipartite cut.

The results improve upon recent bounds but do not resolve the conjectured sharp threshold.

Abstract

The paper is devoted to sufficient conditions for the existence of vertex cuts in simple graphs, where the induced subgraph on the cut vertices belongs to a specified graph class. In particular, we show that any connected graph with $n$ vertices and fewer than $(19n - 28)/8$ edges admits a forest cut. This result improves upon recent bounds, although it does not resolve the conjecture that the sharp threshold is $3n - 6$ (Chernyshev, Rauch, Rautenbach, 2024). Furthermore, we prove that if the number of edges is less than $(80n-134)/31$, then the graph admits a bipartite cut.