Graph-Like Compacta: Characterizations and Eulerian Loops

TL;DR

This paper introduces new characterizations of compact graph-like spaces, linking them to continua and Freudenthal compactifications, and applies these to characterize Eulerian graph-like compacta.

Contribution

It provides novel characterizations of compact graph-like spaces and advances understanding of Eulerian properties in these spaces.

Findings

New characterizations of compact graph-like spaces

Connections to continua and Freudenthal compactifications

Criteria for Eulerian graph-like compacta

Abstract

A compact graph-like space is a triple $(X,V,E)$ where $X$ is a compact, metrizable space, $V \subseteq X$ is a closed zero-dimensional subset, and $E$ is an index set such that $X \setminus V \cong E \times (0,1)$. New characterizations of compact graph-like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied to characterize Eulerian graph-like compacta.