On the graph condition regarding the $F$-inverse cover problem

TL;DR

This paper characterizes the class of graphs related to the $F$-inverse cover problem for finite inverse monoids by identifying forbidden minors for Abelian group varieties, advancing understanding of the algebraic-graph theoretical connection.

Contribution

It proves that the class of graphs satisfying the property for a given group variety is minor-closed and identifies forbidden minors for Abelian groups.

Findings

The class of graphs with the property is minor-closed.

Forbidden minors are explicitly characterized for Abelian groups.

Provides a complete description of graphs related to Abelian group varieties.

Abstract

In their paper titled "On $F$-inverse covers of inverse monoids", Auinger and Szendrei have shown that every finite inverse monoid has an $F$-inverse cover if and only if each finite graph admits a locally finite group variety with a certain property. We study this property and prove that the class of graphs for which a given group variety has the required property is closed downwards in the minor ordering, and can therefore be described by forbidden minors. We find these forbidden minors for all varieties of Abelian groups, thus describing the graphs for which such a group variety satisfies the above mentioned condition.