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On forest and bipartite cuts in sparse graphs

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

TL;DR

The paper investigates vertex cuts whose induced subgraph on the cut-vertices belongs to a fixed class (forest or bipartite) in sparse connected graphs. It develops a linear-programming inspired framework with a quality function $q_{\alpha,\beta}$ and analyzes minimal counterexamples via graph separations to obtain structural constraints. The main contributions are new edge-count bounds, showing that $|E| < \frac{19n-28}{8}$ guarantees a forest cut and $|E| < \frac{80n-134}{31}$ guarantees a bipartite cut, improving prior results though not reaching the conjectured $3n-6$ threshold. The method combines modularity arguments for separations with detailed local-structure lemmas on degree-4 vertices and yields contradiction arguments through carefully constructed inequality systems. These results advance separator theory in sparse graphs and provide a framework that could guide future refinements or extensions to other target classes.

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.

On forest and bipartite cuts in sparse graphs

TL;DR

The paper investigates vertex cuts whose induced subgraph on the cut-vertices belongs to a fixed class (forest or bipartite) in sparse connected graphs. It develops a linear-programming inspired framework with a quality function and analyzes minimal counterexamples via graph separations to obtain structural constraints. The main contributions are new edge-count bounds, showing that guarantees a forest cut and guarantees a bipartite cut, improving prior results though not reaching the conjectured threshold. The method combines modularity arguments for separations with detailed local-structure lemmas on degree-4 vertices and yields contradiction arguments through carefully constructed inequality systems. These results advance separator theory in sparse graphs and provide a framework that could guide future refinements or extensions to other target classes.

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 vertices and fewer than edges admits a forest cut. This result improves upon recent bounds, although it does not resolve the conjecture that the sharp threshold is (Chernyshev, Rauch, Rautenbach, 2024). Furthermore, we prove that if the number of edges is less than , then the graph admits a bipartite cut.

Paper Structure

This paper contains 7 sections, 30 theorems, 46 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of cuts and separations
  4. Properties of the minimal counterexample
  5. Improving the estimate for forest cuts
  6. Estimate for bipartite cuts
  7. Conclusion

Key Result

Theorem 1

Every graph on $n$ vertices with at most $2n-4$ edges has an independent cut.

Figures (6)

  • Figure 1: Three graphs that will appear as induced subgraphs on the cut
  • Figure 2: Shortening the path in Lemma \ref{['cut_lemma_1']}. The cut vertices are colored red.
  • Figure 3: Connected components in Lemma \ref{['cut_lemma_2']}.
  • Figure 4: Cycle on the vertices of the cut in Lemma \ref{['no_c4']}.
  • Figure 5: Neighborhoods of $u$ and $v$ in Lemma \ref{['tree_common_tetrahedron']}.
  • ...and 1 more figures

Theorems & Definitions (64)

  • Theorem 1: G. Chen, X. Yu, 2002
  • Conjecture 1: Chernyshev, Rauch, Rautenbach, 2024
  • Theorem 2
  • Conjecture 2
  • Theorem 3
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1: "Modularity of the quality function"
  • ...and 54 more