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The graph minor relation satisfies the twin alternative conjecture

Jorge Bruno

Abstract

In 2006 Bonato and Tardif posed the Tree Alternative Conjecture (TAC): the equivalence class of a tree under the embeddability relation is, up to isomorphism, either trivial or infinite. In 2022 LaFlamme, et al. provided a rigorous exposition of a conter-example to TAC developed by Tetano in his 2008 PhD thesis. Also in 2022, the present author provided a positive answer to TAC for the topological minor relation. Along with embeddability and the topological minor, the graph minor relation completes the triad of the most widely studied graph relations. In this paper we provide a positive answer to TAC for the the graph minor.

The graph minor relation satisfies the twin alternative conjecture

Abstract

In 2006 Bonato and Tardif posed the Tree Alternative Conjecture (TAC): the equivalence class of a tree under the embeddability relation is, up to isomorphism, either trivial or infinite. In 2022 LaFlamme, et al. provided a rigorous exposition of a conter-example to TAC developed by Tetano in his 2008 PhD thesis. Also in 2022, the present author provided a positive answer to TAC for the topological minor relation. Along with embeddability and the topological minor, the graph minor relation completes the triad of the most widely studied graph relations. In this paper we provide a positive answer to TAC for the the graph minor.
Paper Structure (7 sections, 10 theorems, 9 equations, 1 table)

This paper contains 7 sections, 10 theorems, 9 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Large Trees
  4. Small Trees
  5. RootedTAC($\equiv^*$)
  6. Unrooted
  7. Open Questions

Key Result

Corollary 1.1

For any $\hat{R} \in \{\equiv, \equiv^\sharp, \equiv^*\}$, RootedTAC($\hat{R})$ holds. Further, when considering only small trees, TAC($\hat{R})$ also holds.

Theorems & Definitions (16)

  • Corollary 1.1
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.2
  • proof
  • ...and 6 more