Ann Neudauer
Pacific University
  (Title:) Representations of Bicircular Matroids

  (Abstract:)
  Matroids are everywhere.  Vector spaces are matroids.
Matroids are useful in situations that are modelled
by both graphs and matrices.

We define two matroids on a graph.
Let $G$ be a graph (loops and parallel edges allowed)
with vertex set $V=\{1,2,\ldots,n\}$ and edge set $E$.
In the classical matroid associated with a graph, a set of edges
is independent in the matroid if it
contains no cycles, and the circuits of the matroid are the
single cycles of $G$.  The {\it bicircular matroid} of $G$ is the matroid
$B(G)$ defined on $E$ whose circuits are the
subgraphs which are subdivisions of one of the graphs: (i)
two loops on the same vertex, (ii) two loops joined by an edge, (iii) three
edges joining the same pair of vertices.

The bicircular matroid is known to be a transversal matroid
and thus can be represented by a family of sets, called a presentation.
Given a transversal matroid, we can determine whether it is bicircular.
I will show that, given any presentation of a bicircular matroid,
we can find a graph representing the matroid, and
that, in some cases, there is more than one graph.
We investigate how the graphs representing
a bicircular matroid are related, and what this means in terms of
their presentations as transversal matroids.

What else might we determine about these matroids?
We consider other current problems on matroids.