Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficiently distinguishing all tangles in locally finite graphs

Raphael W. Jacobs, Paul Knappe

Abstract

While finite graphs have tree-decompositions that efficiently distinguish all their tangles, locally finite graphs with thick ends need not have such tree-decompositions. We show that every locally finite graph without thick ends admits such a tree-decomposition, in fact a canonical one. Our proof exhibits a thick end at any obstruction to the existence of such tree-decompositions and builds on new methods for the analysis of the limit behaviour of strictly increasing sequences of separations.

Efficiently distinguishing all tangles in locally finite graphs

Abstract

While finite graphs have tree-decompositions that efficiently distinguish all their tangles, locally finite graphs with thick ends need not have such tree-decompositions. We show that every locally finite graph without thick ends admits such a tree-decomposition, in fact a canonical one. Our proof exhibits a thick end at any obstruction to the existence of such tree-decompositions and builds on new methods for the analysis of the limit behaviour of strictly increasing sequences of separations.
Paper Structure (9 sections, 29 theorems, 1 equation, 2 figures)

This paper contains 9 sections, 29 theorems, 1 equation, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Limit separations
  4. Definition and basic properties
  5. Non-exhaustive sequences and their limit separations
  6. Interplay with the partial order of oriented separations
  7. The interlacing theorem
  8. Proofs of Theorem 1 and 2
  9. Infinitely many rays beyond the limit

Key Result

Theorem 1.1

Every connected finite graph has a canonical tree-decomposition which efficiently distinguishes all its tangles.

Figures (2)

  • Figure 1: A diagram depicting the three sets $X_i$, $Y_i$ and $Z_i$ from the proof of \ref{['lem:FiniteOrderBelowLimit']}.
  • Figure 2: The graph $G$ from \ref{['ex:RaysNotEquivalent']}

Theorems & Definitions (50)

  • Theorem 1.1: ProfilesNew*Theorem 3
  • Theorem 1.2: CanonicalTreesofTDs*Theorem 7.3 & InfiniteSplinters*Theorem 6.6
  • Theorem 1
  • Theorem 2
  • Lemma 2.1: confing*Statement (6)
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4: InfiniteSplinters*Lemma 2.7
  • ...and 40 more