$n$-arc and $n$-circle connected graph-like spaces

TL;DR

This paper investigates the properties of $n$-arc and $n$-circle connectedness in compactifications of locally finite graphs and graph-like continua, revealing notable differences in their connectivity behaviors.

Contribution

It introduces the concepts of $n$-arc and $n$-circle connectedness in the context of graph-like spaces and analyzes their distinct behaviors in different compactifications.

Findings

Distinct behaviors of $n$-arc and $n$-circle connectedness in graph-like spaces

Differences observed between locally finite graph compactifications and continua

New insights into the structure of connected graph-like spaces

Abstract

A space $X$ is $n$-arc connected (respectively, $n$-circle connected) if for any choice of at most $n$ points there is an arc (respectively, a circle) in $X$ containing the specified points. We study $n$-arc connectedness and $n$-circle connectedness in compactifications of locally finite graphs and the slightly more general class of graph-like continua, uncovering a striking difference in their behaviour regarding $n$-arc and -circle connectedness.