Generalizations of Matroid Duality
نویسنده
چکیده
Matroid duality is an important generalization of duality for planar graphs. Using unpublished notes of Brylawski, we extend this notion to arbitrary set systems. This allows one to define a generalized Tutte polynomial. We examine this polynomial for several set systems that are not matroids, and we also investigate the combinatorial significance of duality for these set systems.
منابع مشابه
Generalizations of Wei's Duality Theorem
Wei’s celebrated Duality Theorem is generalized in several ways, expressed as duality theorems for linear codes over division rings and, more generally, duality theorems for matroids. These results are further generalized, resulting in two Wei-type duality theorems for new combinatorial structures that are introduced and named demi-matroids. These generalize matroids and are the appropriate com...
متن کاملOriented Matroid Pairs, Theory and an Electric Application
The property that a pair of oriented matroidsM ? L , M R on E have free union and no common (non-zero) covector generalizes oriented matroid duality. This property characterizes when certain systems of equations whose only nonlinearities occur as real monotone bijections have a unique solution for all values of additive parameters. Instances include sign non-singularity of square matrices and g...
متن کاملOn Brylawski's Generalized Duality
We introduce a notion of duality—due to Brylawski—that generalizes matroid duality to arbitrary rank functions. This allows us to define a generalization of the matroid Tutte polynomial. This polynomial satisfies a deletion-contraction recursion, where deletion and contraction are defined in this more general setting. We explore this notion of duality for greedoids, antimatroids and demi-matroi...
متن کاملWei-type duality theorems for matroids
We present several fundamental duality theorems for matroids and more general combinatorial structures. As a special case, these results show that the maximal cardinalities of fixed-ranked sets of a matroid determine the corresponding maximal cardinalities of the dual matroid. Our main results are applied to perfect matroid designs, graphs, transversals, and linear codes over division rings, in...
متن کاملMatroid Duality From Topological Duality In Surfaces Of Nonnegative Euler Characteristic
One of the most basic examples of matroid duality is the following. Let G be a graph imbedded in the plane and let G∗ be its topological dual graph. If M(G) is the cycle matroid of G, then the dual matroid M∗(G) = M(G∗). If G is a connected graph that is 2-cell imbedded in a surface of demigenus d > 0 (the demigenus is equal to 2 minus the euler characteristic of the surface), then M∗(G) 6= M(G...
متن کامل