How many circuits determine an oriented matroid?

TL;DR

This paper explores the minimal number of circuits required to uniquely determine an oriented matroid, providing bounds and interpretations across various classes including uniform, regular, graphic, and cographic matroids.

Contribution

It introduces new bounds and variants for the circuit determination problem and connects these to concepts in graph theory and design theory.

Findings

Established general upper and lower bounds for connected orientable matroids.

Linked circuit determination to subgraph and intersection graph properties.

Analyzed specific cases for uniform, regular, graphic, and cographic matroids.

Abstract

Las Vergnas and Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present general upper and lower bounds in the setting of general connected orientable matroids, leading to the study of subgraphs of the base graph and the intersection graph of circuits. We then consider the problem for uniform matroids which is closely related to the notion of (connected) covering numbers in Design Theory. Finally, we also devote special attention to regular matroids as well as some graphic and cographic matroids leading in particular to the topics of (connected) bond and cycle covers in Graph Theory.