Finding a small 3-connected minor maintaining a fixed minor and a fixed element

Abstract

LetN andM be 3-connected matroids, whereN is a minor ofM on at least 4 elements, and lete be an element ofM and not ofN. Then, there exists a 3-connected minor\(\bar M\) ofM that usese, hasN as a minor, and has at most 4 elements more thanN. This result generalizes a theorem of Truemper and can be used to prove Seymour’s 2-roundedness theorem, as well as a result of Oxley on triples in nonbinary matroids.

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First author supported by the Alexander von Humboldt Foundation and the Institut für Ökonometrie und Operations Research of the University of Bonn, Federal Republic of Germany, while this author was on leave from the Department of Mathematical Sciences, Rice University, Houston, Texas, USA. Second author supported by a Royal E. Cabell Graduate Fellowship through Northwestern University, Evanston, Illinois, USA.

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Bixby, R.E., Coullard, C.R. Finding a small 3-connected minor maintaining a fixed minor and a fixed element. Combinatorica 7, 231–242 (1987). https://doi.org/10.1007/BF02579300

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AMS subject classification (1980)

  • 05 B 35