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The structure of group-labeled graphs forbidding an immersion

Rose McCarty, Caleb McFarland, Paul Wollan

Abstract

A $Γ$-labeled graph is an oriented graph with edges invertibly labeled by a group $Γ$. We prove a structure theorem for $Γ$-labeled graphs which forbid a fixed $Γ$-labeled graph as an immersion, for any finite $Γ$. Roughly, we show that such graphs admit a tree-cut decomposition in which every bag either contains few high degree vertices or is nearly signed over a proper subgroup of $Γ$.

The structure of group-labeled graphs forbidding an immersion

Abstract

A -labeled graph is an oriented graph with edges invertibly labeled by a group . We prove a structure theorem for -labeled graphs which forbid a fixed -labeled graph as an immersion, for any finite . Roughly, we show that such graphs admit a tree-cut decomposition in which every bag either contains few high degree vertices or is nearly signed over a proper subgroup of .
Paper Structure (8 sections, 16 theorems, 4 equations, 2 figures)

This paper contains 8 sections, 16 theorems, 4 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Immersions
  4. The structure theorem
  5. Packing circuits labeled outside of a subgroup
  6. Finding a rich flower immersion
  7. Proving the structure theorem
  8. Acknowledgments

Key Result

Theorem 1.1

For any finite group $\Gamma$ and any $\Gamma$-labeled graph $(H, \gamma_H)$, there exists an integer $t$ so that for any $2$-edge-connected $\Gamma$-labeled graph $(G, \gamma)$ which forbids an immersion of $(H, \gamma_H)$, there exists a shifting $\gamma'$ of $\gamma$ so that $(G, \gamma')$ "decom

Figures (2)

  • Figure 1: A depiction of an immersion $\varphi$ of $H$ into $G$.
  • Figure 2: A $(3,5)$-flower with one petal highlighted in bold red.

Theorems & Definitions (29)

  • Theorem 1.1: Informal Statement
  • Theorem 2.1
  • Theorem 3.1: PapAPaths
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 19 more