On edge-direction and compact edge-end spaces

TL;DR

The paper develops a parallel theory of edge-direction spaces $\mathcal{D}_E(G)$ to the classical end spaces, establishing a tight link via the line-graph relationship so that $\mathcal{D}_E(G)$ is homeomorphic to $\Omega(G')$. It introduces the completion graph $\tilde{G}$ to realize every edge-direction as an edge-end and proves a key compactness characterization: an edge-end space $\Omega_E(G)$ is compact if and only if it is representable as $\Omega_E(G)\approx\Omega_E(H)\approx\mathcal{D}_E(H)$ for some connected $H$. The work further shows timid vertices capture the essential structure of compact edge-end spaces and proves a full edge-end/timid-end equivalence: the class of timid-end spaces coincides with the class of edge-end spaces, with a robust correspondence between timid-end and timid-direction spaces. Collectively, these results extend Diestel’s directions–ends correspondence to edge-contexts, unify several topological representations (end spaces, edge-end spaces, edge-direction spaces), and provide practical tools (completion graphs and timid frameworks) for analyzing compactness and representation in infinite graphs.

Abstract

Directions of graphs were originally introduced in the study of a cops-and-robbers kind of game, while the study of end spaces has been used to generalize classical graph-theoretical results to infinite graphs, such as Halin's generalization of Menger's theorem. An edge-analogue of end spaces, where finite sets of edges are used instead of vertices as separator agents to form the so-called edge-end space, has been recently used to obtain an edge-analogue of this later result. Inspired by Diestel's correspondence between directions and ends of a graph, we tackle in this paper an edge-analogue of directions, its relation with line graphs, and an edge-analogue of Diestel's correspondence. The results of this study had some implications over edge-end space compactness, which then became a target of inquiry: we thus show an edge-analogue of Diestel's combinatorial characterization for compact end spaces. Non-edge-dominating vertices play an important role in our characterization, which motivated the study of ends and directions using now finite sets of these vertices as separator agents, as done previously for edges, giving rise to other topological spaces associated with graphs. These new direction and end spaces once again motivate an analogue of Diestel's correspondence result, and further generalizations are obtained. All of these constructions define topological space-classes associated with graphs such as edge-end spaces and edge-direction spaces of graphs. The paper organizes these topological space-classes appearing throughout the text with representation results, as it was done by Pitz and Kurkofka, as well as Aurichi, Real and Magalhães Júnior. Most notably, we show that every compact edge-end space can be represented as the edge-direction space of a connected graph.

Figures (9)

• Figure 1.1: For this graph $G$, $\Omega(G)$ has two distinct ends, but $\Omega_E(G)$ has only one edge-end.
• Figure 1.2: A diagram depicting the relationship between the space classes.
• Figure 1.3: For this graph $G$, one has that $\Omega_\mathrm{t}(G)$ has one point, $\Omega_E(G)$ has two and $\Omega(G)$ has three.
• Figure 2.5: Construction of $(e_n^1,e_n^2)$.
• Figure 2.6: For each $e \in F$ we add $\tilde{e}_{e}^i$ in the situation above to $\tilde{F}$, if they exist. This means that $\tilde{e} \notin \tilde{F}\implies e_{\tilde{e}}^i \notin F$.

Theorems & Definitions (79)

• Theorem : Theorem 1.1 in aurichi2024topologicalremarksendedgeend
• Theorem : \ref{['edgedirectionsareendsofedgegraph']}
• Corollary : \ref{['COR_EdgeDirCompact']}
• Corollary : \ref{['directionrepresentation']}
• Corollary : \ref{['raylesscharact']}
• Theorem : \ref{['completiontheorem']}
• Theorem : \ref{['notendspaceofedgegraph']}
• Theorem : Corollary 4.4 in Endspacesandspanningtrees
• Theorem : \ref{['compactnesscharacterization']}
• Theorem : \ref{['THM_CompectEdgeEndDirection']}