Canonical graph decompositions via local separations

TL;DR

This paper develops a new local separation theory to reconstruct canonical graph decompositions from finite, local information, making previously non-computable decompositions effectively computable.

Contribution

It introduces a novel theory of local separations in finite graphs to reconstruct canonical decompositions from finite data, enabling their computability.

Findings

Canonical decompositions are now computable from finite local information.

Develops a new theory of local separations for finite graphs.

Reconstructs global tangle structures using local separations.

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.