Editing Graphic matroid (section)

==Minors and duality==
[[File:Nonisomorphic planar duals.svg|thumb|300px|Two different graphs (red) that are duals of the same planar graph (pale blue). Despite being non-isomorphic as graphs, they have isomorphic graphic matroids.]]
A matroid is graphic if and only if its [[Matroid minor|minors]] do not include any of five forbidden minors: the [[uniform matroid]] <math>U{}^2_4</math>, the [[Fano plane]] or its dual, or the duals of <math>M(K_5)</math> and <math>M(K_{3,3})</math> defined from the [[complete graph]] <math>K_5</math> and the [[complete bipartite graph]] <math>K_{3,3}</math>.<ref name="tutte65">{{citation
 | last = Tutte | first = W. T.
 | journal = Journal of Research of the National Bureau of Standards
 | mr = 0179781
 | pages = 1–47
 | title = Lectures on matroids
 | url = https://nvlpubs.nist.gov/nistpubs/jres/69B/jresv69Bn1-2p1_A1b.pdf
 | volume = 69B
 | year = 1965
 | doi=10.6028/jres.069b.001| doi-access = free
 }}. See in particular section 2.5, "Bond-matroid of a graph", pp. 5–6, section 5.6, "Graphic and co-graphic matroids", pp. 19–20, and section 9, "Graphic matroids", pp. 38–47.</ref><ref>{{citation
 | last = Seymour | first = P. D. | authorlink = Paul Seymour (mathematician)
 | doi = 10.1016/S0167-5060(08)70855-0
 | journal = Annals of Discrete Mathematics
 | mr = 597159
 | pages = 83–90
 | title = On Tutte's characterization of graphic matroids
 | volume = 8
 | year = 1980| isbn = 9780444861108 }}.</ref><ref>{{citation
 | last = Gerards | first = A. M. H.
 | doi = 10.1002/jgt.3190200311
 | issue = 3
 | journal = [[Journal of Graph Theory]]
 | mr = 1355434
 | pages = 351–359
 | title = On Tutte's characterization of graphic matroids—a graphic proof
 | volume = 20
 | year = 1995| s2cid = 31334681
 }}.</ref> The first three of these are the forbidden minors for the regular matroids,<ref>{{citation
 | last = Tutte | first = W. T. | authorlink = W. T. Tutte
 | journal = Transactions of the American Mathematical Society
 | mr = 0101526
 | pages = 144–174
 | title = A homotopy theorem for matroids. I, II
 | volume = 88
 | year = 1958
 | issue = 1 | doi=10.2307/1993244| jstor = 1993244 }}.</ref> and the duals of <math>M(K_5)</math> and <math>M(K_{3,3})</math> are regular but not graphic.

If a matroid is graphic, its dual (a "co-graphic matroid") cannot contain the duals of these five forbidden minors. Thus, the dual must also be regular, and cannot contain as minors the two graphic matroids <math>M(K_5)</math> and <math>M(K_{3,3})</math>.<ref name="tutte65"/>

Because of this characterization and [[Wagner's theorem]] characterizing the [[planar graph]]s as the graphs with no <math>K_5</math> or <math>K_{3,3}</math> [[graph minor]], it follows that a graphic matroid <math>M(G)</math> is co-graphic if and only if <math>G</math> is planar; this is [[Whitney's planarity criterion]]. If <math>G</math> is planar, the dual of <math>M(G)</math> is the graphic matroid of the [[dual graph]] of <math>G</math>. While <math>G</math> may have multiple dual graphs, their graphic matroids are all isomorphic.<ref name="tutte65"/>