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Matroid with complemented basis sets
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
Dual_matroid
Abstraction of linear independence of vectors
In combinatorics, a matroid /ˈmeɪtrɔɪd/ is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many
Matroid
Graph representing faces of another graph
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
Dual_graph
Matroid with graph forests as independent sets
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
Graphic_matroid
Matroid in which every permutation is a symmetry
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
Uniform_matroid
Characterization of planar graphs by matroids
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
Whitney's_planarity_criterion
Matroid with no linear representation
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
Vámos_matroid
General concept and operation in mathematics
matroid theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called the dual matroid.
Duality_(mathematics)
Abstraction of ordered linear algebra
An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane
Oriented_matroid
Abstraction of mod-2 vector independence
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
Binary_matroid
Abstraction of 2-colorable graphs
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
Bipartite_matroid
Matroid that can be represented over all fields
{\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.
Regular_matroid
Join-meet algebra on matroid flats
In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular
Geometric_lattice
Maximum size of an independent set of the 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
Matroid_rank
Independence system partitionable into circuits
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
Eulerian_matroid
Direct sum of uniform matroids
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
Partition_matroid
Matroid obtained by restrictions and contractions
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
Matroid_minor
Vectors with given pattern of independence
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
Matroid_representation
Maximal independent set of the matroid
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
Basis_of_a_matroid
Conjecture on forbidden minors of matroids
{\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
Rota's_conjecture
Abstraction of disjoint paths in directed graphs
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
Gammoid
Largest independent set of paired elements
combinatorial optimization, the matroid parity problem is a problem of finding the largest independent set of paired elements in a matroid, a structure that abstracts
Matroid_parity_problem
Abstraction of algebraic independence
In mathematics, an algebraic matroid is a matroid, a combinatorial structure, that expresses an abstraction of the relation of algebraic independence.
Algebraic_matroid
Abstraction of graph shortest cycles
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
Matroid_girth
Subdivision into few independent sets
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
Matroid_partitioning
Tree which includes all vertices of a graph
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
Spanning_tree
Hierarchical clustering of graph edges
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
Branch-decomposition
Matroid without short circuits
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
Paving_matroid
Geometry with 7 points and 7 lines
structure theory of matroids. Excluding the Fano plane as a matroid minor is necessary to characterize several important classes of matroids, such as regular
Fano_plane
Subroutine for testing independence
mathematics and computer science, a matroid oracle is a subroutine through which an algorithm may access a matroid, an abstract combinatorial structure
Matroid_oracle
Cycle graph plus universal vertex
} In matroid theory, two particularly important special classes of matroids are the wheel matroids and the whirl matroids, both derived from
Wheel_graph
Algebraic encoding of graph connectivity
and number of connected components, with immediate generalizations to matroids. It is also the most general graph invariant that can be defined by a
Tutte_polynomial
Existence of a line through two points
oriented matroid with n {\displaystyle n} elements has at least 3 n / 7 {\displaystyle 3n/7} two-point lines, or equivalently every rank-3 matroid with fewer
Sylvester–Gallai_theorem
Length of a shortest cycle contained in the graph
unified in matroid theory by the girth of a matroid, the size of the smallest dependent set in the matroid. For a graphic matroid, the matroid girth equals
Girth_(graph_theory)
Mathematical structure
mathematics, a base-orderable matroid is a matroid that has the following additional property, related to the bases of the matroid. For any two bases A {\displaystyle
Base-orderable_matroid
Irrational system of points and lines
configuration. This matroid and its dual matroid have been applied in characterizing certain classes of matroids that are linear matroids over the finite
Perles_configuration
Principle in mathematical optimization
Lagrangian dual problem but other dual problems are used – for example, the Wolfe dual problem and the Fenchel dual problem. The Lagrangian dual problem
Duality_(optimization)
Graph which remains connected when fewer than k edges are removed
edge connectivity of its dual graph, and vice versa. These concepts are unified in matroid theory by the girth of a matroid, the size of the smallest
Edge_connectivity
Fewest graph edges whose removal breaks all cycles
dimension of the cycle space of the graph, in terms of matroid theory as the dual rank of its graphic matroid, and in terms of topology as one of the Betti numbers
Cyclomatic_number
Matroid theory
Matroid-constrained number partitioning is a variant of the multiway number partitioning problem, in which the subsets in the partition should be independent
Matroid-constrained number partitioning
Matroid-constrained_number_partitioning
British-Canadian codebreaker and mathematician (1917–2002)
graphic matroid. The algorithm makes use of the fact that a planar graph is simply a graph whose circuit-matroid, the dual of its bond-matroid, is graphic
W._T._Tutte
lattice and corresponds to a matroid of finite rank. Semimodular lattices are also known as upper semimodular lattices; the dual notion is that of a lower
Semimodular_lattice
Method to solve optimization problems
Method of computing optimal strategies for last-success problems Oriented matroid – Abstraction of ordered linear algebra Quadratic programming – Solving
Linear_programming
Points separated from others by a line
can be generalized to one of parametric optimization in a matroid: one is given a matroid in which each element is weighted by a linear function of a
K-set_(geometry)
Geometric structure of 8 points and 8 lines
as a matroid, whose elements are the points of the configuration and whose nontrivial flats are the lines of the configuration. In this matroid, a set
Möbius–Kantor_configuration
Austrian American mathematician
Brigitte Irma Servatius (born 1954) is a mathematician specializing in matroids and structural rigidity. She is a professor of mathematics at Worcester
Brigitte_Servatius
Set that intersects every one of a family of sets
finite sets form the basis sets of a matroid, the transversal matroid of C. The independent sets of the transversal matroid are the partial transversals of
Transversal_(combinatorics)
Method for mathematical optimization
on his previous papers on oriented-matroid theory. However, Bland's rule exhibits cycling on some oriented-matroid linear-programming problems. The first
Criss-cross_algorithm
Number of forests a graph's edges may be partitioned into
special case of a more general matroid partitioning problem, in which one wishes to express a set of elements of a matroid as a union of a small number
Arboricity
American/Canadian mathematician and computer scientist
he proved the matroid intersection theorem, a very general combinatorial min-max theorem which, in modern terms, showed that the matroid intersection problem
Jack_Edmonds
Topics referred to by the same term
function Basis (linear algebra) Dual basis Orthonormal basis Schauder basis Basis (universal algebra) Basis of a matroid Generating set of an ideal: Gröbner
Basis
Textbook on the theory of matroids
using circuits, matroid rank, and submodular set function are also presented, as are sums, minors, truncations, and duals of matroids. Chapter three concerns
Independence Theory in Combinatorics
Independence_Theory_in_Combinatorics
Pseudolines arranged largely to study arrangements of lines
flip graph. Each rank-3 oriented matroid is equivalent to an arrangement of pseudolines, and each oriented matroid which is also uniform (in which the
Arrangement_of_pseudolines
or a base of this matroid. Cardinality constraints are special cases of matroid constraints in which the matroid is a uniform matroid. Categorized cardinality
Balanced_number_partitioning
Maximal subgraph whose vertices can reach each other
{\displaystyle n-c} is the matroid-theoretic rank of the graph, and the rank of its graphic matroid. The rank of the dual cographic matroid equals the circuit
Component_(graph_theory)
Mathematical operator
A and {x}. A finitary closure operator with this property is called a matroid. The dimension of a vector space, or the transcendence degree of a field
Closure_operator
linear and affine Gale diagrams can also be described through the duality of oriented matroids. As with the linear diagram, a subset of vertices forms a face
Gale_diagram
Pencil and paper connection game
the Shannon switching game played on a directed graph and an oriented matroid have been described for theoretical purposes; but no corresponding commercial
Shannon_switching_game
Japanese mathematician (born 1951)
his contributions to optimization, polyhedral computation and oriented matroid theory. Fukuda is a professor in optimization and computational geometry
Komei_Fukuda
Software for the algorithmic treatment of convex polyhedra
programmatically TropLi: for computing tropical linear spaces of matroids tosimplex: Dual simplex algorithm implemented by Thomas Opfer Vinci: volumes of
Polymake
into the Matroid and mechas' colliding attacks, which causes a time warp that sends Alata and Bakutofuji-ER back in time and shrinks the Matroid. They eventually
List of Tensou Sentai Goseiger characters
List_of_Tensou_Sentai_Goseiger_characters
Set whose pairs have minima and maxima
algebras, Boolean algebras, distributive lattices, and geometric lattices (matroids). These lattice-like structures all admit order-theoretic as well as algebraic
Lattice_(order)
The Avis–Fukuda algorithm adapted the criss-cross algorithm for oriented matroids. A 2025 article by Zelin Dong, Fenglei Fan, Huan Xiong, and Tieyong Zeng
Vertex_enumeration_problem
Generalization of graph theory
abstract simplicial complex with the augmentation property is called a matroid. Laminar: for any two hyperedges, either they are disjoint, or one is included
Hypergraph
Result in combinatorics and graph theory
to determine the existence of a transversal which is independent in a matroid. Hall 1986, pg. 51. An alternative form of the marriage theorem applies
Hall's_marriage_theorem
Mathematical system of orderings or sets
defining antimatroids as set systems are very similar to those of matroids, but whereas matroids are defined by an exchange axiom, antimatroids are defined instead
Antimatroid
Indian-American mathematician
schemes defined by Kirchhoff polynomials to the representation spaces of matroids. Moreover, using Mnev's universality theorem, we show that these schemes
Prakash_Belkale
Element of graph theory
ISBN 978-0-521-59840-8, MR 1477750. Las Vergnas, Michel (1980), "Convexity in oriented matroids", Journal of Combinatorial Theory, Series B, 29 (2): 231–243, doi:10
Acyclic_orientation
American mathematician (born 1934)
seminal. The rational singularity and fundamental cycles, which are used in matroid theory, are such examples of his sheer originality and thinking. He began
Michael_Artin
Affine subspace of a Euclidean space
example Dihedral angle (between two planes). See also Angles between flats.) Matroid Coplanarity Isometry Gallier, J. (2011). "Basics of Affine Geometry". Geometric
Flat_(geometry)
Japanese voice actor
Shinkenger: Kusare Ayakashi Azemidoro (ep. 31) Tensou Sentai Goseiger: Matroid Bakutofuji-ER of the Timer (ep. 39–40) Tokumei Sentai Go-Busters: Omochiloid
Kōichi_Sakaguchi
Cycles in a graph that generate all cycles
weight of its longest cycle. In any vector space, and more generally in any matroid, a minimum weight basis may be found by a greedy algorithm that considers
Cycle_basis
Flat-sided three-dimensional shape
Bokowski, J.; Guedes de Oliveira, A. (2000), "On the generation of oriented matroids", Discrete and Computational Geometry, 24 (2–3): 197–208, doi:10.1007/s004540010027
Polyhedron
of it include enumerative combinatorics, combinatorial design theory, matroid theory, extremal combinatorics and algebraic combinatorics, as well as
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Geometric system of two mutually inscribed tetrahedra
the two configurations, including the fact that both are self-dual under Matroid duality. In abstract terms, the latter configuration has "points" 0,
Möbius_configuration
Equivalence of optimization problems
of Max-Flow Min-Cut Theorem". Combinatorial Optimization: Networks and Matroids. Dover. pp. 117–120. ISBN 0-486-41453-1. Christos H. Papadimitriou, Kenneth
Max-flow_min-cut_theorem
American mathematician
theorem Convex cone Duality (mathematics) Monotone operator (Cyclic decomposition of maximal monotone operator) Oriented matroids (realizable OMs and
R._Tyrrell_Rockafellar
Convex polyhedron projected from hypercube
addition, certain Catalan solids (duals of Archimedean solids) are again zonohedra: Kepler's rhombic dodecahedron is the dual of the cuboctahedron. The rhombic
Zonohedron
11613, doi:10.1016/j.jcta.2020.105273, S2CID 51775423 Luoto, K. (2008), "A matroid-friendly basis for the quasisymmetric functions", Journal of Combinatorial
Quasisymmetric_function
"Cocircuit Graphs and Efficient Orientation Reconstruction in Oriented Matroids". Eur. J. Comb. 22 (5): 587–600. doi:10.1006/eujc.2001.0481. ISSN 0195-6698
Graph_of_a_polytope
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
are invertible over the field. Balanced matrix Regular matroid Special linear group Total dual integrality Hermite normal form The term was coined by
Unimodular_matrix
Toroidal polyhedron with 14 triangle faces
Bokowski, J.; Guedes de Oliveira, A. (2000), "On the Generation of Oriented Matroids", Discrete & Computational Geometry, 24: 197–208, doi:10.1007/s004540010027
Császár_polyhedron
1112/blms/18.6.571, MR 0859948 Ramírez Alfonsín, J. L. (2001), "Lawrence oriented matroids and a problem of McMullen on projective equivalences of polytopes", European
McMullen_problem
Algorithm that outputs all solutions to a problem
the Bron–Kerbosch algorithm Listing all elements of structures such as matroids and greedoids Several problems on graphs, e.g., enumerating independent
Enumeration_algorithm
analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal
Lists_of_mathematics_topics
Graded lattice with modular maximal chain
supersolvable, although it is not geometric. The lattice of flats of the graphic matroid for a graph is supersolvable if and only if the graph is chordal. Working
Supersolvable_lattice
Vertices connected in pairs by edges
they allow for higher-dimensional simplices. Every graph gives rise to a matroid. In model theory, a graph is just a structure. But in that case, there
Graph_(discrete_mathematics)
Smallest convex set containing a given set
convex hulls may also be generalized in a more abstract way, to oriented matroids. It is not obvious that the first definition makes sense: why should there
Convex_hull
Type of random graph
"The multivariate Tutte polynomial (Alias Potts model) for graphs and matroids". Surveys in Combinatorics 2005. pp. 173–226. arXiv:math/0503607. doi:10
Random_cluster_model
the graphic matroid of a graph, a subset of edges is independent if the corresponding subgraph is a tree or forest. In the bicircular matroid, a subset
Glossary_of_graph_theory
British mathematician
York, 1994 Borovik, Alexandre V.; Gelfand, I. M.; White, Neil: Coxeter matroids. Progress in Mathematics, 216. Birkhäuser Boston, Inc., Boston, MA, 2003
Alexandre_Borovik
Russian-British mathematician
first to recognize the importance of transversal matroids, and he showed that transversal matroids can be represented using linear algebra over transcendental
Leon_Mirsky
Social choice problem
Munagala and Shah focus on three types of constraints: Matroid constraints: there is a fixed matroid M over the items, and the chosen items must form a basis
Multi-issue_voting
Combinitorics of Polyhedra
facets are available. Abstract polytope Combinatorial commutative algebra Matroid polytope Order polytope Simplicial sphere Stable matching polytope Ziegler
Polyhedral_combinatorics
Economical computational problem
the bases of a matroid over the set of resources, then all best-response sequences converge in polynomial number of steps, and the matroid property is essential
Nash_equilibrium_computation
Problem of finding the longest simple path for a given graph
ISBN 9780262032933. Lawler, Eugene L. (2001), Combinatorial Optimization: Networks and Matroids, Courier Dover Publications, p. 64, ISBN 9780486414539. Sedgewick, Robert;
Longest_path_problem
Soviet mathematician (1913–2009)
MR 2000133 Borovik, Alexandre V.; Gelfand, I. M.; White, Neil (2003), Coxeter matroids, Progress in Mathematics, vol. 216, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3764-4
Israel_Gelfand
Hungarian mathematician (born 1955)
independently published on the criss-cross algorithm. The theory of oriented matroids has also been used by Terlaky and Zhang (1991) to prove that their criss-cross
Tamás_Terlaky
Concerned with the notion of stability in model theory
e. is prime and minimal over) a strongly minimal set, which carries a matroid structure determined by (model-theoretic) algebraic closure that gives
Stable_theory
Describing a family of graphs by excluding certain (sub)graphs
finite obstruction set. Erdős–Hajnal conjecture Forbidden subgraph problem Matroid minor Zarankiewicz problem Diestel, Reinhard (2000), Graph Theory, Graduate
Forbidden graph characterization
Forbidden_graph_characterization
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