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Two characterisations of accessible quasi-transitive graphs

Matthias Hamann, Babak Miraftab

Abstract

We prove two characterisations of accessibility of locally finite quasi-transitive connected graphs. First, we prove that any such graph $G$ is accessible if and only if its set of separations of finite order is an ${\rm Aut}(G)$-finitely generated semiring. The second characterisation says that $G$ is accessible if and only if every process of splittings in terms of tree amalgamations stops after finitely many steps.

Two characterisations of accessible quasi-transitive graphs

Abstract

We prove two characterisations of accessibility of locally finite quasi-transitive connected graphs. First, we prove that any such graph is accessible if and only if its set of separations of finite order is an -finitely generated semiring. The second characterisation says that is accessible if and only if every process of splittings in terms of tree amalgamations stops after finitely many steps.

Paper Structure

This paper contains 5 sections, 16 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Tree amalgamations
  3. The semiring $\mathcal{S}(G)$
  4. Iterated splittings
  5. The main theorem

Key Result

Theorem 1.1

HLMR Every multi-ended quasi-tran-si-tive locally finite connected graph is a non-trivial tree amalgamation of two quasi-tran-si-tive locally finite connected graphs of finite adhesion and finite identification, distinguishing ends and respecting the action of the involved groups.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • Proposition 3.4
  • proof
  • Lemma 4.1
  • proof
  • ...and 18 more