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arXiv · 2411.17885

Extremal Problems on Forest Cuts and Acyclic Neighborhoods in Sparse Graphs

F. Botler·Y. S. Couto·C. G. Fernandes·E. F. de Figueiredo·R. Gómez·V. F. dos Santos·C. M. Sato
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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.

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F. Botler, Y. S. Couto, C. G. Fernandes, E. F. de Figueiredo, R. Gómez, V. F. dos Santos, C. M. Sato. 2024-11-26. Extremal Problems on Forest Cuts and Acyclic Neighborhoods in Sparse Graphs. https://arxiv.org/abs/2411.17885

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