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arXiv · 2505.16179

On forest and bipartite cuts in sparse graphs

Ilya I. Bogdanov·Elizaveta Neustroeva·Georgy Sokolov·Alexei Volostnov·Nikolay Russkin·Vsevolod Voronov
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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.

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Ilya I. Bogdanov, Elizaveta Neustroeva, Georgy Sokolov, Alexei Volostnov, Nikolay Russkin, Vsevolod Voronov. 2025-05-22. On forest and bipartite cuts in sparse graphs. https://arxiv.org/abs/2505.16179

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