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arXiv · 2501.16170

Canonical graph decompositions via local separations

Raphael W. Jacobs·Paul Knappe·Jan Kurkofka
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Abstract

Every finite graph $G$ can be decomposed in a canonical way that displays its local connectivity-structure [DJKK26]. These decompositions are defined via a suitable more tree-like covering of $G$, whose tangle-tree structure is projected down to $G$. The covering graphs needed here are almost always infinite, and their tangle-tree structure is defined in terms of their (global) low-order separations. The canonical decompositions they induce on $G$ are therefore not computable following their definition. We reconstruct these decompositions of $G$ from finite information in $G$ itself that is sufficiently local to be reflected in the cover. This involves the reconstruction of canonical tangle structure in terms of a new theory of local separations in finite graphs, which we develop for this purpose. As an application, we find that the canonical graph-decompositions from [DJKK26] are computable.

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Raphael W. Jacobs, Paul Knappe, Jan Kurkofka. 2025-01-27. Canonical graph decompositions via local separations. https://arxiv.org/abs/2501.16170

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