n-Arc Connected Graphs

TL;DR

This paper provides a complete classification of n-arc connected graphs by characterizing their topological and combinatorial properties, revealing equivalences and conditions for various levels of arc connectivity.

Contribution

It introduces a comprehensive classification of n-arc connected graphs, linking topological properties with graph subdivisions and connectivity conditions, extending previous partial results.

Findings

|G| is n-ac for all n if and only if G is a subdivision of one of nine graphs.

|G| is 6-ac if G is one of nine graphs or 3-regular, 3-connected, with certain edge removal properties.

Complete classification of n-ac graphs for all n is achieved.

Abstract

Given a graph G, of arbitrary size and unbounded vertex degree, denote by |G| the one-complex associated with $G$. The topological space |G| is n-arc connected (n-ac) if every set of no more than n points of |G| are contained in an arc (a homeomorphic copy of the closed unit interval). For any graph G, we show the following are equivalent: (i) |G| in 7-ac, (ii) |G| is n-ac for all n, and (iii) G is a subdivision of one of nine graphs. A graph G has |G| 6-ac if and only if either G is one of the nine 7-ac graphs, or, after suppressing all degree-2-vertices, the graph G is 3-regular, 3-connected, and removing any 6 edges does not disconnect G into 4 or more components. Similar combinatorial characterizations of graphs G such that |G| is n-ac for n=3, 4 and 5 are given. Together these results yield a complete classification of n-ac graphs, for all n.