Geodetic topological cycles in locally finite graphs

TL;DR

This paper proves that in locally finite graphs, the topological cycle space is generated by geodetic topological circles, highlighting differences between finite and geodetic cycles in generating the cycle space.

Contribution

It establishes that geodetic topological circles generate the topological cycle space in locally finite graphs, a novel insight into their structure.

Findings

Geodetic topological circles generate the cycle space in locally finite graphs.

Finite cycles generate the cycle space, but finite geodetic cycles may not.

Finite geodetic cycles do not necessarily generate the cycle space.

Abstract

We prove that the topological cycle space C(G) of a locally finite graph G is generated by its geodetic topological circles. We further show that, although the finite cycles of G generate C(G), its finite geodetic cycles need not generate C(G).