In matroid theory, the dual of a matroid M {\displaystyle M} is another matroid M ∗ {\displaystyle M^{\ast }} that has the same elements as M {\displaystyle...

In combinatorics, a matroid /ˈmeɪtrɔɪd/ is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many...

finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids. A matroid that is both graphic and co-graphic...

by the concept of a dual matroid. Variations of planar graph duality include a version of duality for directed graphs, and duality for graphs embedded...

In matroid theory, an Eulerian matroid is a matroid whose elements can be partitioned into a collection of disjoint circuits. In a uniform matroid U n...

planar if and only if its graphic matroid is also cographic (that is, it is the dual matroid of another graphic matroid). In purely graph-theoretic terms...

matroid theory, a binary matroid is a matroid that can be represented over the finite field GF(2). That is, up to isomorphism, they are the matroids whose...

matroid theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called the dual matroid....

Vámos matroid is a paving matroid, meaning that all of its circuits have size at least equal to its rank. The Vámos matroid is isomorphic to its dual matroid...

theory of matroids, the rank of a matroid is the maximum size of an independent set in the matroid. The rank of a subset S of elements of the matroid is, similarly...

elements. The matroid U n 2 {\displaystyle U{}_{n}^{2}} is called the n {\displaystyle n} -point line. The dual matroid of the uniform matroid U n r {\displaystyle...

In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular...

{\displaystyle F} . If a matroid is regular, so is its dual matroid, and so is every one of its minors. Every direct sum of regular matroids remains regular....

An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane...

In matroid theory, a field within mathematics, a gammoid is a certain kind of matroid, describing sets of vertices that can be reached by vertex-disjoint...

theory of matroids, a matroid representation is a family of vectors whose linear independence relation is the same as that of a given matroid. Matroid representations...

In mathematics, a basis of a matroid is a maximal independent set of the matroid—that is, an independent set that is not contained in any other independent...

In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which...

In mathematics, an algebraic matroid is a matroid, a combinatorial structure, that expresses an abstraction of the relation of algebraic independence....

matroid is bipartite if and only if its dual matroid is an Eulerian matroid, a matroid that can be partitioned into disjoint circuits. For matroids that...

or dependent set. The cogirth of a matroid is the girth of its dual matroid. Matroid girth generalizes the notion of the shortest cycle in a graph, the...

{\displaystyle n/2} two-point lines. The dual of the non-Fano matroid. The eight-point matroid of a square antiprism. The matroid obtained by relaxing the unique...

combinatorial optimization, the matroid parity problem is a problem of finding the largest independent set of paired elements in a matroid. The problem was formulated...

of matroids, a minor of a matroid M is another matroid N that is obtained from M by a sequence of restriction and contraction operations. Matroid minors...

cutset. This duality can also be expressed using the theory of matroids, according to which a spanning tree is a base of the graphic matroid, a fundamental...

Matroid partitioning is a problem arising in the mathematical study of matroids and in the design and analysis of algorithms. Its goal is to partition...

branchwidth of its associated graphic matroid. The branchwidth of a matroid is equal to the branchwidth of its dual matroid, and in particular this implies...

mathematical theory of matroids, a paving matroid is a matroid in which every circuit has size at least as large as the matroid's rank. In a matroid of rank r {\displaystyle...

mathematics and computer science, a matroid oracle is a subroutine through which an algorithm may access a matroid, an abstract combinatorial structure...

matrix whose columns are the v i {\displaystyle v_{i}} . Then the vector matroid M _ {\displaystyle {\underline {\mathcal {M}}}} on the columns of M {\displaystyle...