A sparsity matroid is a mathematical structure that captures how densely a multigraph is populated with edges. To unpack this a little, sparsity is a measure of density of a graph that bounds the number of edges in any subgraph. The property of having a particular matroid as its density measure is invariant under graph isomorphisms and so it is a graph invariant.
The graphs we are concerned with generalise simple directed graphs by allowing multiple same-oriented edges between pairs of vertices. Matroids are a quite general mathematical abstraction that describe the amount of indepdendence in, variously, points in geometric space and paths in a graph; when applied to characterising sparsity, matroids describe certain sets of sparse graphs. These matroids are connected to the structural rigidity of graphs and their ability to be decomposed into edge-disjoint spanning trees via the Tutte and Nash-Williams theorem. There is a family of efficient algorithms, known as pebble games, for determining if a multigraph meets the given sparsity condition.
A sparsitymatroid is a mathematical structure that captures how densely a multigraph is populated with edges. To unpack this a little, sparsity is a measure...
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...
In matroid theory, a mathematical discipline, the girth of a matroid is the size of its smallest circuit or dependent set. The cogirth of a matroid is...
many edges as they have vertices) – and their matroid structure allows several other families of sparse graphs to be decomposed as unions of forests and...
In the mathematics of structural rigidity, a rigidity matroid is a matroid that describes the number of degrees of freedom of an undirected graph with...
mathematics concerning the study of finite or countable discrete structures. Matroid Greedoid Ramsey theory Van der Waerden's theorem Hales–Jewett theorem Umbral...
Ramsey Theory (Jaroslav Nešetřil and Vojtěch Rödl, eds., 1990, vol. 5) Matroid Theory and its Applications in Electric Network Theory and in Statics (Andras...
rigid graphs, and they form the bases of the two-dimensional rigidity matroids. If n points in the plane are given, then there are 2n degrees of freedom...
given size and connected components, with immediate generalizations to matroids. It is also the most general graph invariant that can be defined by a...
programming Odds algorithm used to solve optimal stopping problems Oriented matroid Quadratic programming, a superset of linear programming Semidefinite programming...
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...
mathematical structures other than graphs, and in particular in vector spaces and matroids. Two algorithmic problems are associated with MISs: finding a single MIS...
graph H. Graph minors are often studied in the more general context of matroid minors. In this context, it is common to assume that all graphs are connected...
large linear-programming problems A is typically a sparse matrix and, when the resulting sparsity of B is exploited when maintaining its invertible representation...
abstract simplicial complex with the augmentation property is called a matroid. Laminar: for any two hyperedges, either they are disjoint, or one is included...
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...
1017/CBO9780511662119.010, MR 1477750. Las Vergnas, Michel (1980), "Convexity in oriented matroids", Journal of Combinatorial Theory, Series B, 29 (2): 231–243, doi:10...
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...
reduction can in fact be performed as the complex is constructed by using matroid theory, leading to further performance increases. Another recent algorithm...
RLNC-based protocols such as tunable sparse network coding that exploit this knowledge. These protocols introduce sparsity (zero-elements) in the coding coefficients...
mathematician specializing in disease modeling Collette Coullard, American matroid theorist and operations researcher Judith Covington, American mathematics...
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