Explore: Oriented Matroids

independent sets). The distinction between matroids and oriented matroids is discussed further below. Matroids are often useful in areas such as dimension

about oriented matroids, a good preparation is to study the textbook on linear optimization by Nering and Tucker, which is infused with oriented-matroid ideas

M(G)} is called a cycle matroid. Matroids derived in this way are graphic matroids. Not every matroid is graphic, but all matroids on three elements are

Dr. Lukas Finschi, "Homepage of Oriented Matroids" Handbook of Discrete and Computational Geometry Arrangement of lines Oriented matroid Coxeter group

for oriented matroids. The criss-cross algorithm has been adapted for problems that are more complicated than linear programming: There are oriented-matroid

Bernd; White, Neil; Ziegler, Günter (1999). "10 Linear programming". Oriented Matroids. Cambridge University Press. pp. 417–479. doi:10.1017/CBO9780511586507

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

number of ordinary lines can be generalized to oriented matroids: every rank-3 oriented matroid with n {\displaystyle n} elements has at least 3 n / 7

cocircuits of M {\displaystyle {\mathcal {M}}} and if we consider the oriented matroid M {\displaystyle {\mathcal {M}}} represented by M {\displaystyle {M}}

represent algebraic (or semialgebraic) varieties as realization spaces of oriented matroids. Informally it can also be understood as the statement that point