# Two matroids on graphs: How do they compare?

## Document Type

Presentation Abstract

10-4-2010

## Abstract

In 1933, three Harvard junior-fellows tied together some recurring themes in mathematics, into what Gian Carlo Rota called one of the most important ideas of our day. They were finding independence everywhere they looked. Do you? We find that matroids are everywhere: Vector spaces are matroids; We can define matroids on a graph. Matroids are useful in situations that are modelled by both graphs and matrices. Bicircular matroids model generalized network flow problems whose algorithms are more efficient than those available for general linear programming codes.

Two matroids are commonly defined on a graph: the familiar cycle matroid and the more rarely-encountered bicircular matroid. In the cycle matroid, a set of edges is independent in the matroid if it contains no cycles in the graph, and the circuits of the matroid are the single cycles of the graph. In the bicircular matroid, two cycles in the graph form a circuit of the matroid. More specifically, the circuits are the subgraphs which are subdivisions of one of the following graphs: (i) two loops on the same vertex, (ii) two loops joined by an edge, (iii) three edges joining the same pair of vertices.

What questions can we ask about matroids and what might we count? We will discuss some recent results for bicircular matroids.