We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.

Given an undirected graph G = (V,E) and a delta-matroid (V,F), the delta-matroid matching problem is to find a maximum cardinality matching M such that the set of the end vertices of M belongs to F . This problem is a natural generalization of the matroid matching problem to delta-matroids, and thus it cannot be solved in polynomial time in general. This paper introduces a class of the delta-ma...

AND GENERIC RIGIDITY IN THE PLANE SACHIN PATKAR, BRIGITTE SERVATIUS, AND K. V. SUBRAHMANYAM Abstract. We consider the concept of abstract 2–dimensional rigidity and provide necessary and sufficient conditions for a matroid to be an abstract rigidity matroid of a complete graph. This characterization is a natural extenWe consider the concept of abstract 2–dimensional rigidity and provide necessa...

The matroid parity (MP) problem is a natural extension of the matching problem to the matroid setting. It can be formulated as a 0− 1 linear program using the so-called rank and line constraints. We call the associated family of polytopes MP polytopes. We then prove the following: (i) when the matroid is a gammoid, each MP polytope is a projection of a perfect matching polytope into a suitable ...

We give an example of a simple oriented matroid D that admits an oriented adjoint. Already any adjoint of the underlying matroid D, however, does itself not admit an adjoint. D arises from the wellknown Non-Desargues-Matroid by a coextension by a coparallel element and, hence, has rank 4. The orientability of D and some of its adjoints follows from an apparantly new oriented matroid constructio...

In this note, we study a constrained independent set problem for matroids and certain generalizations. The basic problem can be regarded as an ordered version of the matroid parity problem. By a reduction of this problem to matroid intersection, we prove a min-max formula. Studying the weighted case and a delta-matroid generalization, we prove that some of them are not more complex than matroid...

We discuss the relationship between matroid rank functions and a concept of discrete concavity called M-concavity. It is known that a matroid rank function and its weighted version called a weighted rank function are M-concave functions, while the (weighted) sum of matroid rank functions is not M-concave in general. We present a sufficient condition for a weighted sum of matroid rank functions ...