0. The Flow Lattice of Oriented Matroids
Winfried Hochstättler, Robert Nickel
Mathematical Foundations of Computer Science
Department of Mathematics
Brandenburg Technical University Cottbus

1. Outline Circuits and flows Reorientation and geometry
An approach to define a flow number of an oriented matroid
The flow lattice
of regular oriented matroids
of rank 3 oriented matroids
of uniform oriented matroids
Outlook

2. Circuits and Flows in Digraphs
Directed circuit :
Circular flow
Flow number: 1 -1 2 -2 1 -1 1 4 3 5 7 2 8 6 9

3. Points in the Space
Radon’s Theorem (1952): Let pairwise different. For all with exists a partition (Radon partition), so that
Such a partition implies a signing of the elements 1 2 3 S T 5 4 6 7
Generalization of circuits in loop-less digraphs

4. Reorientation and Geometry
digraphs are reoriented by flipping edges
for point configurations we need some projective geometry:
put the points on the projective sphere
reorientation of an element is done by replacing the point by its double on the opposite half sphere
a projective transformation then defines a new equator so that all points are on one half sphere (see Grünbaum – Convex Polytopes) 1 2 3 4 5
Circuits:
1234:+--+0
1235:+--0+
1245:+-0-+
1345:+0+-+
2345:0++-+
Circuits reor.:
1234:+-++0
1235:+-+0+
1345:+0--+
2345:0+--+

5. Circuits and Flows of an Oriented Matroid
Let be a family of signed subsets of a finite set that satisfies the following conditions:
Then is the set of signed circuits of an oriented matroid
A flow in is an integer combination of signed characteristic vectors of circuits:

6. A Flow Number for Oriented Matroids
Goddyn, Tarsi, Zhang 1998: Let be the set of co-circuits of and the set of all reorientations. Then the oriented flow number is defined as
For graphic matroids equal to the circular flow number of the graph (involves Hoffman’s Circulation Lemma 1960: for each bond in the digraph)
Rank 3: (M. Edmonds, McNulty 2004)
General case (co-connected): (Goddyn, Hliněný, Hochstättler)

7. Not a Matroid Invariant
Different orientations of the same underlying matroid (e. g ) can lead to a different oriented flow number - + - +

8. The Flow Lattice The flow lattice of an oriented matroid is defined as
We define a flow number of analog to the flow number of a digraph
What is the dimension of ?
Does have a short characterization?
Does contain a basis of ?
Determine the flow number!

9. Regular Oriented Matroids (Digraphs and more)
Concerning the dimension of we have
is regular
The elementary circuits to a basis of form a basis of
The computation of is known to be an hard problem
For digraphs:
Tutte’s 6-flow theorem
Tutte’s 3-, 4-, 5-flow conjectures

10. Rank 3 (Points in the plane)
Let be non-uniform (uniform case considered later)
Theorem: Any connected co-simple non-uniform oriented matroid of rank 3 with more than 6 elements has trivial flow lattice (i. e ).
co-simple means (for rank 3): does not contain an point line
is the maximum regular oriented matroid of rank 3
The flow number is 2
A basis of is constructed inductively

11. The Uniform Case (Points in General Position)
points do not share a hyperplane
Any circuit has elements
Example:

12. The Uniform Case (Points in General Position)
For even rank (odd dimension) we have: (Hochstättler, Nešetřil 2003)
Theorem: Let be a uniform oriented matroid of odd rank on elements Then if and only if there is a reorientation ( ) so that
There is a reorientation with balanced circuits:
is a neighborly matroid polytope

13. The Uniform Case (Points in General Position)
Theorem (structure of the lattice):
Flow number:
Basis construction: Let If is neighborly for all then is neighborly, too.
Construct the basis inductively.

14. Summary
Let be simple and co-simple on more than 6 elements

15. Outlook
Does any (rank-preserving) single element extension of a (maximal) regular oriented matroid increase the dimension by ?
What is the dimension of for general oriented matroids?
Does always have a basis of signed circuits?
Is there an orientable matroid so that but ?
Otherwise would be well defined for orientable matroids.

16. Thanks for your attention