David Forge

Incremental construction of alpha lattices and association rules

In this paper we discuss Alpha Galois lattices (Alpha lattices for short) and the corresponding association rules. An alpha lattice is coarser than the related concept lattice and so contains fewer nodes, so fewer closed patterns, and a smaller basis of association rules. Coarseness depends on a a priori classification, i.e. a cover C of the powerset of the instance set I, and on a granularity parameter α. In this paper, we define and experiment a Merge operator that when applied to two Alpha lattices G(C1, α) and G(C2, α) generates the Alpha lattice G(C1∪C2, α), so leading to a class-incremental construction of Alpha lattices. We then briefly discuss the implementation of the incremental process and describe the min-max bases of association rules extracted from Alpha lattices.

Bijections between affine hyperplane arrangements and valued graphs

We show new bijective proofs of previously known formulas for the number of regions of some deformations of the braid arrangement, by means of a bijection between the no-broken-circuit sets of the corresponding integral gain graphs and some kinds of labelled binary trees. This leads to new bijective proofs for the Shi, Catalan, and similar hyperplane arrangements.

Minimal non-orientable matroids in a projective plane

We construct a new family of minimal non-orientable matroids of rank three. Some of these matroids embed in Desarguesian projective planes. This answers a question of Ziegler: for every prime power q, find a minimal non-orientable submatroid of the projective plane over the q-element field.

How is a chordal graph like a supersolvable binary matroid

Let G be a finite simple graph. From the pioneering work of R.P. Stanley it is known that the cycle matroid of G is supersolvable iff G is chordal (rigid): this is another way to read Diracs theorem on chordal graphs. Chordal binary matroids are in general not supersolvable. Nevertheless we prove that, for every supersolvable binary matroid M, a maximal chain of modular flats of M canonically determines a chordal graph.

The directed switching game on Lawrence oriented matroids

The main content of the note is a proof of the conjecture of Hamidoune and Las Vergnas on the directed switching game in the case of Lawrence oriented matroids.

A note on Tutte polynomials and Orlik--Solomon algebras

Let A¢ = {H1,....,Hn} be a (central) arrangement of hyperplanes in ¢d and M(A¢) the dependence matroid of the linear forms {θHi ∈ (¢d)* : Ker(θHi) = Hi}. The Orlik-Solomon algebra OS(M) of a matroid M is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The graded algebra OS(M(A¢)) is isomorphic to the cohomology algebra of the manifold m = ¢d\H ∈ A¢ H. The Tutte polynomial TM(x, y) is a powerful invariant of the matroid M. When M(A¢) is a rank 3 matroid and the θHi are complexifications of real linear forms, we will prove that OS(M) determines TM(x, y). This result partially solves a conjecture of Falk.

Lattice points in orthotopes and a huge polynomial Tutte invariant of weighted gain graphs

A gain graph is a graph whose edges are orientably labeled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy Tuttes deletion-contraction and multiplicative identities). In order to do that we develop a relative of the Tutte polynomial of a semimatroid. Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavskys two for gain graphs, Noble and Welshs for graphs with positive integer weights, and that of rooted integral gain graphs by Forge and Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that remains to be found.An evaluation of one example of our polynomial counts proper list colorations of the gain graph from a color set with a gain-group action. When the gain group is Z d , the lists are order ideals in the integer lattice Z d , and there are specified upper bounds on the colors, then there is a formula for the number of bounded proper colorations that is a piecewise polynomial function of the upper bounds, of degree nd where n is the order of the graph.This example leads to graph-theoretical formulas for the number of integer lattice points in an orthotope but outside a finite number of affinographic hyperplanes, and for the number of n � d integral matrices that lie between two specified matrices but not in any of certain subspaces defined by simple row equations.

Coverings of the Vertices of a Graph by Small Cycles

Given a graph G with n vertices, we call ck(G) the minimum number of elementary cycles of length at most k necessary to cover the vertices of G. We bound ck(G) from the minimum degree and the order of the graph.