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Canonical graph decompositions and local separations: From infinite coverings to a finite combinatorial theory

Johannes Carmesin, Raphael W. Jacobs, Paul Knappe, Jan Kurkofka

TL;DR

This work develops a finite, combinatorial theory of r-local graph decompositions that faithfully encode local connectivity via r-local coverings. By introducing r-local separations, bottlenecks, and a displacement parameter Δ_r(G), the authors build canonical graph-decompositions H_r^{≤k}(G) that capture all r-local bottlenecks up to order k and are computable by a two-step algorithm. The theory harmonizes with existing local coverings, provides lifting/projection tools, and extends the Tree-of-Tangles paradigm to a local-bottlenecks setting, enabling a finite, algorithmic handle on graphs with complex local structure. In particular, the results unify local and global perspectives, recover trees-of-tangles in special cases, and offer a practical path to finite descriptions of otherwise infinite coverings, with potential applications to sparse networks and group-related graph structure.

Abstract

Every finite graph $G$ can be decomposed in a canonical way that displays its local connectivity-structure [DJKK22]. For this, they consider a suitable covering of $G$, an inherently infinite concept from Topology, and project its tangle-tree structure to $G$. We present a construction of these decompositions via a finite combinatorial theory of local separations, which we introduce here and which is compatible with the covering-perspective.

Canonical graph decompositions and local separations: From infinite coverings to a finite combinatorial theory

TL;DR

This work develops a finite, combinatorial theory of r-local graph decompositions that faithfully encode local connectivity via r-local coverings. By introducing r-local separations, bottlenecks, and a displacement parameter Δ_r(G), the authors build canonical graph-decompositions H_r^{≤k}(G) that capture all r-local bottlenecks up to order k and are computable by a two-step algorithm. The theory harmonizes with existing local coverings, provides lifting/projection tools, and extends the Tree-of-Tangles paradigm to a local-bottlenecks setting, enabling a finite, algorithmic handle on graphs with complex local structure. In particular, the results unify local and global perspectives, recover trees-of-tangles in special cases, and offer a practical path to finite descriptions of otherwise infinite coverings, with potential applications to sparse networks and group-related graph structure.

Abstract

Every finite graph can be decomposed in a canonical way that displays its local connectivity-structure [DJKK22]. For this, they consider a suitable covering of , an inherently infinite concept from Topology, and project its tangle-tree structure to . We present a construction of these decompositions via a finite combinatorial theory of local separations, which we introduce here and which is compatible with the covering-perspective.

Paper Structure

This paper contains 35 sections, 54 theorems, 20 equations, 17 figures.

Table of Contents

  1. Introduction
  2. From tree-decompositions to graph-decompositions
  3. From separations to local separations
  4. Graph-decompositions and local separations
  5. From trees of tangles to graphs of local bottlenecks
  6. Organisation of the paper
  7. Acknowledgements
  8. Review of graph-decompositions and coverings
  9. Local coverings
  10. Separations
  11. Decompositions
  12. Bottlenecks
  13. Bottlenecks
  14. A tree of bottlenecks
  15. Local separations
  16. ...and 20 more sections

Key Result

Theorem 1

Let $G$ be a finite connected graph and $r,k \in \mathbb N$ with $k<K(G,r)$.

Figures (17)

  • Figure 1: A tree-decomposition of a graph (black) into $K_5$'s (grey).
  • Figure 2: Graphs whose connectivity-structure eludes trees-of-tangles
  • Figure 3: Graph-decompositions of black graphs into smaller parts (grey). The decomposition graphs are blue.
  • Figure 4: An $H$-decomposition of $G$ (black) into 4-cycles and a $K_5$ (grey) over a cycle $H$ (blue), obtained from a tree-decomposition of the $r$-local cover $G_r$ for $r=4$.
  • Figure 5: Local separators (red) that separate the reddish balls around them.
  • ...and 12 more figures

Theorems & Definitions (152)

  • Theorem 1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Example 3.1
  • Lemma 3.2
  • proof
  • ...and 142 more