Families of matroids induced by classes of graphs

Journal article

It is easily proved that, if P is a class of graphs that is closed under induced subgraphs, then the family of matroids whose basis graphs belong to P is closed under minors. We give simple necessary and sufficient conditions for a minor-closed class of matroids to be induced in this way, and characterise when such a class of matroids contains arbitrarily large connected matroids. We show that five easily-defined families of matroids can be induced by a class of graphs in this manner: binary matroids; regular matroids; the polygon matroids of planar graphs; those matroids for which every connected component is either graphic or cographic; and those matroids for which every connected component is either binary or can be obtained from a binary matroid by a single circuit-hyperplane relaxation. We give an excluded-minor characterisation of the penultimate class, and show that the last of these classes has infinitely many excluded minors.

Authors: Mayhew, D

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Advances in Applied Mathematics
• Volume: 34
• Issue: 3
• Pages: 616–633
• Publication date: 2005-04-01
• Edition: Publisher's version
• DOI: 10.1016/j.aam.2004.11.001
• ISSN: 0196-8858
• Language: English
• Keywords: Rank-preserving weak map; Hereditary class; Excluded-minor characterisation; Matroid basis graphs
• Subjects: Mathematics