An approximate cactus-decomposition for vertex-connectivity
Abstract
The classical cactus theorem of Dinitz, Karzanov and Lomonosov from 1976 states that the minimum edge-cuts of a connected graph are displayed by a cactus. For vertex-separations, Tutte's theorem gives an exact cactus-decomposition for connectivity $2$, but exact cactus-decompositions fail for higher connectivity. Extending earlier approximate cactus-decomposition results for connectivities $3$ and $4$ by Carmesin and Kurkofka [FOCS 2023] and Kurkofka and Planken [SODA 2026] to all connectivities, we prove that for every k∈ℕ there is a constant d=d(k) such that every k-connected graph G admits a cactus-decomposition displaying k-separations such that every k-separation of G has edit-distance at most d to a displayed separation. Our proof develops tools that may be useful beyond this result. The first is a three-dimensional version of the classical corner-diagram method, which we use to show that triples of crossing k-separations have a controlled two-dimensional behaviour. The second is a covering-space approach for studying local crossing patterns from a global perspective. Third, we advance the theory of flowers, which originated in matroid theory.
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BibTeX
@misc{bourneuf2026approximate,
title = {{An approximate cactus-decomposition for vertex-connectivity}},
author = {Romain Bourneuf and Johannes Carmesin and Jan Kurkofka and Tim Planken},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01772},
eprint = {2611.01772},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01772},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An approximate cactus-decomposition for vertex-connectivity
AU - Romain Bourneuf
AU - Johannes Carmesin
AU - Jan Kurkofka
AU - Tim Planken
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01772
UR - https://arxiv.org/abs/2611.01772
AB - The classical cactus theorem of Dinitz, Karzanov and Lomonosov from 1976 states that the minimum edge-cuts of a connected graph are displayed by a cactus. For vertex-separations, Tutte's theorem gives an exact cactus-decomposition for connectivity $2$, but exact cactus-decompositions fail for higher connectivity. Extending earlier approximate cactus-decomposition results for connectivities $3$ and $4$ by Carmesin and Kurkofka [FOCS 2023] and Kurkofka and Planken [SODA 2026] to all connectivities, we prove that for every k∈ℕ there is a constant d=d(k) such that every k-connected graph G admits a cactus-decomposition displaying k-separations such that every k-separation of G has edit-distance at most d to a displayed separation. Our proof develops tools that may be useful beyond this result. The first is a three-dimensional version of the classical corner-diagram method, which we use to show that triples of crossing k-separations have a controlled two-dimensional behaviour. The second is a covering-space approach for studying local crossing patterns from a global perspective. Third, we advance the theory of flowers, which originated in matroid theory.
ER -