Countably determined ends and graphs

TL;DR

This paper characterizes the structure of infinite graph ends, called directions, using star-like substructures, rays, and tree-decompositions, revealing how these directions can be distinguished or not by countable choices.

Contribution

It provides a structural characterization of directions in infinite graphs and their countably determined variants using normal trees and tree-decompositions.

Findings

Characterization of directions not uniquely determined by countable choices.

Graphs with all directions countably determined but indistinguishable by countable choices.

Structural criteria for graphs whose directions can be distinguished by countably many choices.

Abstract

The directions of an infinite graph $G$ are a tangle-like description of its ends: they are choice functions that choose compatibly for all finite vertex sets $X\subseteq V(G)$ a component of $G-X$. Although every direction is induced by a ray, there exist directions of graphs that are not uniquely determined by any countable subset of their choices. We characterise these directions and their countably determined counterparts in terms of star-like substructures or rays of the graph. Curiously, there exist graphs whose directions are all countably determined but which cannot be distinguished all at once by countably many choices. We structurally characterise the graphs whose directions can be distinguished all at once by countably many choices, and we structurally characterise the graphs which admit no such countably many choices. Our characterisations are phrased in terms of normal trees and tree-decompositions. Our four (sub)structural characterisations imply combinatorial characterisations of the four classes of infinite graphs that are defined by the first and second axiom of countability applied to their end spaces: the two classes of graphs whose end spaces are first countable or second countable, respectively, and the complements of these two classes.