Kol Kom 04 The Flow Lattice of Oriented

Kol. Kom 04 The Flow Lattice of Oriented Matroids Winfried Hochstättler, Robert Nickel Mathematical Foundations of Computer Science Department of Mathematics Brandenburg Technical University Cottbus

Outline Kol. Kom 04 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 Slide 1 of 16

Circuits and Flows in Digraphs Kol. Kom 04 Directed circuit 1 4 1 1 3 2 1 -2 -1 Circular flow 5 1 7 1 8 1 6 12 9 -1 Flow number: Slide 2 of 16 :

Points in the Space Kol. Kom 04 Radon’s Theorem (1952): Let with (Radon partition), so that pairwise different. For all exists a partition Such a partition implies a signing of the elements 1 2 3 5 4 T S 6 7 Slide 3 of 16

Reorientation and Geometry Kol. Kom 04 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: +--+0 1235: +--0+ 1245: +-0 -+ 1345: +0+-+ 2345: 0++-+ 1 3 2 4 5 Slide 4 of 16 Circuits reor. : 1234: +-++0 1235: +-+0+ 1245: +-0 -+ 1345: +0 --+ 2345: 0+--+

Kol. Kom 04 Circuits and Flows of an Oriented Matroid Let be a family of signed subsets of a finite set following conditions: that satisfies the Then is the set of signed circuits of an oriented matroid A flow in is an integer combination of signed characteristic vectors of circuits: Slide 5 of 16

Kol. Kom 04 A Flow Number for Oriented Matroids Goddyn, Tarsi, Zhang 1998: Let be the set of co-circuits of and Then the oriented flow number is defined as the set of all reorientations. 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, Mc. Nulty 2004) General case (co-connected): (Goddyn, Hliněný, Hochstättler) Slide 6 of 16

Not a Matroid Invariant Kol. Kom 04 Different orientations of the same underlying matroid (e. g. to a different oriented flow number + + - - - + + + - Slide 7 of 16 + ) can lead

The Flow Lattice Kol. Kom 04 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! Slide 8 of 16

Kol. Kom 04 Regular Oriented Matroids (Digraphs and more) Concerning the dimension of we have is regular The elementary circuits The computation of to a basis is known to be an For digraphs: Tutte’s 6 -flow theorem Tutte’s 3 -, 4 -, 5 -flow conjectures Slide 9 of 16 of form a basis of -hard problem

Rank 3 (Points in the plane) Kol. Kom 04 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 is the maximum regular oriented matroid of rank 3 The flow number is 2 A basis of is constructed inductively Slide 10 of 16 -point line

Kol. Kom 04 The Uniform Case (Points in General Position) points do not share a hyperplane Any circuit has elements Example: Slide 11 of 16

The Uniform Case (Points in General Position) Kol. Kom 04 For even rank (odd dimension) we have: (Hochstättler, Nešetřil 2003) Theorem: Let Then so that be a uniform oriented matroid of odd rank on if and only if there is a reorientation ( There is a reorientation with balanced circuits: is a neighborly matroid polytope Slide 12 of 16 elements. )

Kol. Kom 04 The Uniform Case (Points in General Position) Theorem (structure of the lattice): Flow number: Basis construction: Let If is neighborly for all Construct the basis inductively. then Slide 13 of 16 is neighborly, too.

Summary Kol. Kom 04 Let be simple and co-simple on more than 6 elements Slide 14 of 16

Outlook Kol. Kom 04 Does any (rank-preserving) single element extension of a (maximal) regular oriented matroid increase the dimension by ? What is the dimension of Does for general oriented matroids? always have a basis of signed circuits? Is there an orientable matroid so that but Otherwise ? would be well defined for orientable matroids. Slide 15 of 16

Kol. Kom 04 Thanks for your attention Slide 16 of 16