In the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the forests in a given finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids.[1] A matroid that is both graphic and co-graphic is sometimes called a planar matroid (but this should not be confused with matroids of rank 3, which generalize planar point configurations); these are exactly the graphic matroids formed from planar graphs.
^Tutte (1965) uses a reversed terminology, in which he called bond matroids "graphic" and cycle matroids "co-graphic", but this has not been followed by later authors.
In the mathematical theory of matroids, a graphicmatroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the...
cycle matroid. Matroids derived in this way are graphicmatroids. Not every matroid is graphic, but all matroids on three elements are graphic. Every...
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...
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...
basis has a specialized name in several specialized kinds of matroids: In a graphicmatroid, where the independent sets are the forests, the bases are called...
the lattice of partitions corresponds to the lattice of flats of the graphicmatroid of the complete graph. Another example illustrates refinement of partitions...
dimension of the cycle space of a graph, in terms of matroid theory as the corank of a graphicmatroid, and in terms of topology as one of the Betti numbers...
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...
In mathematics, a bipartite matroid is a matroid all of whose circuits have even size. A uniform matroid U n r {\displaystyle U{}_{n}^{r}} is bipartite...
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...
combinatorial problems. For instance, the girth of a co-graphicmatroid (or the cogirth of a graphicmatroid) equals the edge connectivity of the underlying graph...
transversal matroid and a strict gammoid. Not every uniform matroid is graphic, and the uniform matroids provide the smallest example of a non-graphicmatroid, U...
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 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...
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...
graphic matroid (and every co-graphicmatroid) is regular. Conversely, every regular matroid may be constructed by combining graphicmatroids, co-graphic matroids...
structure from which the matroid was defined for graphicmatroids, transversal matroids, gammoids, and linear matroids, and for matroids formed from these by...
derived from wheel graphs. The k-wheel matroid is the graphicmatroid of a wheel Wk+1, while the k-whirl matroid is derived from the k-wheel by considering...
also be expressed using the theory of matroids, according to which a spanning tree is a base of the graphicmatroid, a fundamental cycle is the unique circuit...
bridge, a two-edge path) but the graphicmatroid formed by this bridge is not connected, so no circuit of the graphicmatroid of K 2 , 3 {\displaystyle K_{2...
the matroid intersection problem is to find a largest common independent set in two matroids over the same ground set. If the elements of the matroid are...
Global Information