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The Flow Lattice of Oriented Matroids

Winfried Hochstättler, Robert Nickel

Mathematical Foundations of Computer Science

Department of Mathematics

Brandenburg Technical University Cottbus

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KolKom04Outline

Circuits and flows

Reorientation and geometry

An approach to define a flow number of an oriented matroid

The flow latticeof regular oriented matroids

of rank 3 oriented matroids

of uniform oriented matroids

Outlook

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KolKom04Circuits and Flows in Digraphs

Directed circuit :

Circular flow

Flow number:

2

3

4

5

6

7

8

9

1

1

-11

-1

1

1

1

1

-12

-21 1 1

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Such a partition implies a signing of the elements

Points in the Space

Radon’s Theorem (1952): Let pairwise different. For all with exists a partition (Radon partition), so that

S T

1 2

3

45

67

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KolKom04Reorientation 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)

Circuits:1234:+--+01235:+--0+1245:+-0-+1345:+0+-+2345:0++-+

Circuits reor.:1234:+-++01235:+-+0+1245:+-0-+1345:+0--+2345:0+--+

1

2

3

4

5

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KolKom04Circuits 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:

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KolKom04A 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)

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KolKom04Not a Matroid Invariant

Different orientations of the same underlying matroid (e. g. ) can lead to a different oriented flow number

-

+

++

-

-

-

+

-

++

-

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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!

The Flow Lattice

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KolKom04Regular 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

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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

Rank 3 (Points in the plane)

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The Uniform Case (Points in General Position)

points do not share a hyperplane

Any circuit has elements

Example:

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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

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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.

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KolKom04Summary

Let be simple and co-simple on more than 6 elements

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KolKom04Outlook

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.

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Thanks for your attention