page_title: Graphic matroid
section_title: Minors and duality
hierarchical_section_title: Graphic matroid / Minors and duality
context_page_description: In the mathematical theory of matroids, a graphic 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. A matroid that is both graphic and co-graphic is called a planar matroid; these are exactly the graphic matroids formed from planar graphs.
context_section_description: A matroid is graphic if and only if its minors do not include any of five forbidden minors: the uniform matroid , the Fano plane or its dual, or the duals of  and  defined from the complete graph  and the complete bipartite graph . The first three of these are the forbidden minors for the regular matroids, and the duals of  and  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  and .
  Because of this characterization and Wagner's theorem characterizing the planar graphs as the graphs with no  or  graph minor, it follows that a graphic matroid  is co-graphic if and only if  is planar; this is Whitney's planarity criterion. If  is planar, the dual of  is the graphic matroid of the dual graph of . While  may have multiple dual graphs, their graphic matroids are all isomorphic.
caption: 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.