The Coming of the Matroids

Of course, if we take S to be the set of columns of a matrix, and the independent sets to be the ones that are linearly independent, we get a first example, called a linear matroid. Another important class consists of the graphic ones – here S is the set of edges of a graph G and a subset is independent if it forms a forest. Whitney established some equivalent versions of the axioms, highlighted the above two examples, and proved several basic results. In particular, he showed that, given a matroid M , one gets a second dual matroid M∗ by declaring independent all the sets whose deletion from S do not lower its rank. This generalizes the notion of duality in planar graphs. In addition, he observed that the rank function r satisfies what we now call the submodular property: For all subsets A,B of S