Editing Graphic matroid (section)

==Related classes of matroids==
Some classes of matroid have been defined from well-known families of graphs, by phrasing a characterization of these graphs in terms that make sense more generally for matroids. These include the [[bipartite matroid]]s, in which every circuit is even, and the [[Eulerian matroid]]s, which can be partitioned into disjoint circuits. A graphic matroid is bipartite if and only if it comes from a [[bipartite graph]] and a graphic matroid is Eulerian if and only if it comes from an [[Eulerian graph]]. Within the graphic matroids (and more generally within the [[binary matroid]]s) these two classes are dual: a graphic matroid is bipartite if and only if its [[dual matroid]] is Eulerian, and a graphic matroid is Eulerian if and only if its dual matroid is bipartite.<ref name="w69">{{citation
 | last = Welsh | first = D. J. A. | authorlink = Dominic Welsh
 | journal = [[Journal of Combinatorial Theory]]
 | mr = 0237368
 | pages = 375–377
 | title = Euler and bipartite matroids
 | volume = 6
 | year = 1969
 | issue = 4 | doi=10.1016/s0021-9800(69)80033-5
 | doi-access = free
 }}.</ref>

Graphic matroids are one-dimensional [[rigidity matroid]]s, matroids describing the degrees of freedom of structures of rigid beams that can rotate freely at the vertices where they meet. In one dimension, such a structure has a number of degrees of freedom equal to its number of connected components (the number of vertices minus the matroid rank) and in higher dimensions the number of degrees of freedom of a ''d''-dimensional structure with ''n'' vertices is ''dn'' minus the matroid rank. In two-dimensional rigidity matroids, the [[Laman graph]]s play the role that spanning trees play in graphic matroids, but the structure of rigidity matroids in dimensions greater than two is not well understood.<ref name="whiteley">{{citation
 | last = Whiteley | first = Walter | authorlink = Walter Whiteley
 | contribution = Some matroids from discrete applied geometry
 | doi = 10.1090/conm/197/02540
 | location = Providence, RI
 | mr = 1411692
 | pages = 171–311
 | publisher = American Mathematical Society
 | series = Contemporary Mathematics
 | title = Matroid theory (Seattle, WA, 1995)
 | volume = 197
 | year = 1996| doi-access = free
 | isbn = 978-0-8218-0508-4 }}.</ref>