Fixing numbers for matroids

TL;DR

This paper introduces the fixing number for matroids, establishing bounds and exploring its properties, especially for cycle and bicircular matroids of 3-connected graphs, with connections to permutation groups.

Contribution

It defines the fixing number for matroids and provides bounds, along with new results relating fixing numbers of specific matroids and their automorphism groups.

Findings

Bounds for fixing numbers in terms of size and orbit size

Fixing numbers for cycle and bicircular matroids of 3-connected graphs are equal

Connections between fixing numbers and permutation group actions

Abstract

Motivated by work in graph theory, we define the fixing number for a matroid. We give upper and lower bounds for fixing numbers for a general matroid in terms of the size and maximum orbit size (under the action of the matroid automorphism group). We prove the fixing numbers for the cycle matroid and bicircular matroid associated with 3-connected graphs are identical. Many of these results have interpretations through permutation groups, and we make this connection explicit.