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Edge-ends versus topological ends of graphs

Leandro Aurichi, Paulo Magalhães Júnior, Guilherme Eduardo Pinto

Abstract

Diestel and Kühn proved that the topological ends of an infinite graph are precisely its undominated graph ends, yielding a canonical embedding of the space of topological ends into the space of graph ends. For edge-ends, introduced by Hahn, Laviolette and Širáň, such an embedding does not exist in general. In this note, we characterize the class of infinite graphs for which the topological ends admit a natural injective map into the space of edge-ends that is compatible with the canonical maps between end spaces. Our characterization is purely combinatorial and is expressed in terms of edge-equivalence classes of vertices. Moreover, when such an embedding exists, we identify precisely which edge-ends arise from topological ends, showing that they are exactly the edge-ends containing a non-dominated ray. This establishes a parallel result to the theorem of Diestel and Kühn for edge-end spaces.

Edge-ends versus topological ends of graphs

Abstract

Diestel and Kühn proved that the topological ends of an infinite graph are precisely its undominated graph ends, yielding a canonical embedding of the space of topological ends into the space of graph ends. For edge-ends, introduced by Hahn, Laviolette and Širáň, such an embedding does not exist in general. In this note, we characterize the class of infinite graphs for which the topological ends admit a natural injective map into the space of edge-ends that is compatible with the canonical maps between end spaces. Our characterization is purely combinatorial and is expressed in terms of edge-equivalence classes of vertices. Moreover, when such an embedding exists, we identify precisely which edge-ends arise from topological ends, showing that they are exactly the edge-ends containing a non-dominated ray. This establishes a parallel result to the theorem of Diestel and Kühn for edge-end spaces.
Paper Structure (2 sections, 3 theorems, 2 equations, 4 figures)

This paper contains 2 sections, 3 theorems, 2 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The characterization

Key Result

Lemma 2.5

Let $U$ be an infinite set of vertices in a connected graph $G$. Then $G$ contains either a comb attached to $U$ or a star attached to $U$.

Figures (4)

  • Figure 1: A graph where it is not possible to obtain an injection from $\Omega'(G)$ into $\Omega_E(G)$.
  • Figure 2: Graph where $f_E$ is injective, but $f$ is not.
  • Figure 3: Representation
  • Figure :

Theorems & Definitions (17)

  • Example 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Lemma 2.5: Star-Comb lemma
  • Theorem 2.6
  • proof
  • Claim 2.7
  • proof
  • ...and 7 more