# Circuits and circulant minors

## TL;DR

This paper characterizes when circular matrices have circulant contraction minors using digraph circuits, providing new necessary and sufficient conditions and an alternative characterization for circulant matrices.

## Contribution

It introduces necessary and sufficient conditions for circulant minors in circular matrices based on associated digraph circuits, offering a novel characterization.

## Key findings

- Provides a complete characterization of circulant minors in circular matrices.
- Offers an alternative proof for circulant matrices case.
- Enhances understanding of ideal circular matrices and their minors.

## Abstract

Circulant contraction minors play a key role for characterizing ideal circular matrices in terms of minimally non ideal structures. In this article we prove necessary and sufficient conditions for a circular matrix $A$ to have circulant contraction minors in terms of circuits in a digraph associated with $A$. In the particular case when $A$ itself is a circulant matrix, our result provides an alternative characterization to the one previously known from the literature.