Strong Arcwise Connectedness

TL;DR

This paper investigates the properties of strong arc connectedness in various spaces, establishing existence, non-existence, and characterizations of n-strong arc connectedness in graphs and continua.

Contribution

It provides new results on the existence and characterization of n-strong arc connectedness in graphs and continua, including the first examples of omega-sac rational continua.

Findings

No 4-sac graphs exist

There are 3-sac graphs and 2-sac but not 3-sac graphs

An omega-sac rational continuum exists

Abstract

A space is `n-strong arc connected' (n-sac) if for any n points in the space there is an arc in the space visiting them in order. A space is omega-strong arc connected (omega-sac) if it is n-sac for all n. We study these properties in finite graphs, regular continua, and rational continua. There are no 4-sac graphs, but there are 3-sac graphs and graphs which are 2-sac but not 3-sac. For every n there is an n-sac regular continuum, but no regular continuum is omega-sac. There is an omega-sac rational continuum. For graphs we give a simple characterization of those graphs which are 3-sac. It is shown, using ideas from descriptive set theory, that there is no simple characterization of n-sac, or omega-sac, rational continua.