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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. Outline. Circuits and flows Reorientation and geometry
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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
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
1 1 -2 -1 1 1 2 1 -1 -1 1 1 1 1 Circuits and Flows in Digraphs • Directed circuit : • Circular flow • Flow number: 4 1 3 5 7 2 8 6 9
T S 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 5 4 6 7 • Generalization of circuits in loop-less digraphs
1 Circuits: 1234:+--+0 1235:+--0+ 1245:+-0-+ 1345:+0+-+ 2345:0++-+ Circuits reor.: 1234:+-++0 1235:+-+0+ 1245:+-0-+ 1345:+0--+ 2345:0+--+ 3 2 4 5 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)
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:
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)
+ + - - - - - + + + + - Not a Matroid Invariant • Different orientations of the same underlying matroid (e. g. ) can lead to a different oriented flow number
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!
Regular Oriented Matroids (Digraphs and more) • Concerning the dimension of we have is regular • The elementary circuits to a basis of form a basisof • The computation of is known to be an -hard problem • For digraphs: • Tutte’s 6-flow theorem • Tutte’s 3-, 4-, 5-flow conjectures
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
The Uniform Case (Points in General Position) • points do not share a hyperplane • Any circuit has elements • Example:
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
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.
Summary • Let be simple and co-simple on more than 6 elements
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.
