Circuit Preserving Edge Maps

TL;DR

This paper proves that any one-to-one edge map between 3-connected graphs that preserves circuits is induced by a vertex isomorphism, generalizing Whitney's classical result to possibly infinite graphs.

Contribution

It extends Whitney's theorem by showing circuit-preserving edge maps are induced by vertex isomorphisms in 3-connected graphs, including infinite cases.

Findings

Circuit-preserving edge maps are induced by vertex isomorphisms.

Generalization of Whitney's theorem to infinite graphs.

Applicable to 3-connected graphs, finite or infinite.

Abstract

It is proved that any one-to-one edge map f from a 3-connected graph G onto a graph H, G and H possibly infinite, satisfying f(C) is a circuit in H whenever C is a circuit in G is induced by a vertex isomorphism. This generalizes a result of Whitney which hypothesizes f(C) is a circuit in H if and only if C is a circuit in G.