Different ways to define a matroid

This week, we discussed four different ways to define a matroid, as well as the operations of deletion and contraction.

We began with the independent set definition. From there, we introduced the concepts of bases, circuits, and rank functions, exploring how a matroid can be defined using each of these structures independently. See also section 1.1-1.3 in Oxley, James, Matroid Theory, 2nd edn, Oxford Graduate Texts in Mathematics (Oxford, 2011; online edn, Oxford Academic, 17 Dec. 2013), https://doi.org/10.1093/acprof:oso/9780198566946.001.0001. All these definitions are equivalent, and each offers a unique perspective on the structure of a matroid. To help solidify these concepts, we walked through a few examples, such as uniform matroids and graphic matroids.

Finally, we introduced the deletion and contraction operations, examining them more comprehensively within the specific context of graphic matroids.