{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:08Z","timestamp":1753893788255,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Knowing the excluded minors for a minor-closed matroid property provides a useful alternative characterization of that property. It has been shown in [R. Hall, J. Oxley, C. Semple, G. Whittle, On Matroids of Branch-Width Three, submitted 2001] that if $M$ is an excluded minor for matroids of branch-width $3$, then $ M$ has at most $14$ elements. We show that there are exactly $10$ such binary matroids $M$ (7 of which are regular), proving a conjecture formulated by Dharmatilake in 1994. We also construct numbers of such ternary and quaternary matroids $ M$, and provide a simple practical algorithm for finding a width-$3$ branch-decomposition of a matroid. The arguments in our paper are computer-assisted \u2014 we use a program $MACEK$ [P. Hlin\u011bn\u00fd, The MACEK Program, http:\/\/www.mcs.vuw.ac.nz\/research\/macek, 2002] for structural computations with represented matroids. Unfortunately, it seems to be infeasible to search through all matroids on at most $14$ elements.<\/jats:p>","DOI":"10.37236\/1648","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:25:30Z","timestamp":1578691530000},"source":"Crossref","is-referenced-by-count":4,"title":["On the Excluded Minors for Matroids of Branch-Width Three"],"prefix":"10.37236","volume":"9","author":[{"given":"Petr","family":"Hlin\u011bn\u00fd","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2002,7,23]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v9i1r32\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v9i1r32\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T00:13:41Z","timestamp":1579306421000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v9i1r32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,7,23]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2002,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1648","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2002,7,23]]},"article-number":"R32"}}