Realizable cycle structures in digraphs

TL;DR

This paper explores the algebraic structure of simple cycles in directed graphs, revealing which properties are preserved or lost under this structure and establishing the decidability of cycle configuration realizability.

Contribution

It demonstrates that many graph properties are not inferable from the cycle monoid and proves that determining cycle configuration realizability is decidable via polynomial equations.

Findings

Most graph properties are not determined by the cycle monoid.

Many cycle configurations are impossible even in multidigraphs.

Cycle realizability is decidable and reduces to solving polynomial systems.

Abstract

Simple cycles on a digraph form a trace monoid under the rule that two such cycles commute if and only if they are vertex disjoint. This rule describes the spatial configuration of simple cycles on the digraph. Cartier and Foata have showed that all combinatorial properties of closed walks are dictated by this trace monoid. We find that most graph properties can be lost while maintaining the monoidal structure of simple cycles and thus cannot be inferred from it, including vertex-transitivity, regularity, planarity, Hamiltonicity, graph spectra, degree distribution and more. Conversely we find that even allowing for multidigraphs, many configurations of simple cycles are not possible at all. The problem of determining whether a given configuration of simple cycles is realizable is highly non-trivial. We show at least that it is decidable and equivalent to the existence of integer solutions to systems of polynomial equations.