{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:43Z","timestamp":1753893823028,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For a matroid $M$ having $m$ rank-one flats, the density $d(M)$ is $\\tfrac{m}{r(M)}$ unless $m = 0$, in which case $d(M)= 0$. A matroid is density-critical if all of its proper minors of non-zero rank have lower density. By a 1965 theorem of Edmonds, a matroid that is minor-minimal among simple matroids that cannot be covered by $k$ independent sets is density-critical. It is straightforward to show that $U_{1,k+1}$ is the only minor-minimal loopless matroid with no covering by $k$ independent sets. We prove that there are exactly ten minor-minimal simple obstructions to a matroid being able to be covered by two independent sets. These ten matroids are precisely the density-critical matroids $M$ such that $d(M) &gt; 2$ but $d(N) \\le 2$ for all proper minors $N$ of $M$. All density-critical matroids of density less than $2$ are series-parallel networks. For $k \\ge 2$, although finding all density-critical matroids of density at most $k$ does not seem straightforward, we do solve this problem for $k=\\tfrac{9}{4}$.\u00a0<\/jats:p>","DOI":"10.37236\/8584","type":"journal-article","created":{"date-parts":[[2020,5,29]],"date-time":"2020-05-29T02:20:10Z","timestamp":1590718810000},"source":"Crossref","is-referenced-by-count":0,"title":["On Density-Critical Matroids"],"prefix":"10.37236","volume":"27","author":[{"given":"Rutger","family":"Campbell","sequence":"first","affiliation":[]},{"given":"Kevin","family":"Grace","sequence":"additional","affiliation":[]},{"given":"James","family":"Oxley","sequence":"additional","affiliation":[]},{"given":"Geoff","family":"Whittle","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2020,5,29]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p35\/8092","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p35\/8092","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,29]],"date-time":"2020-05-29T02:20:10Z","timestamp":1590718810000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i2p35"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,29]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,4,3]]}},"URL":"https:\/\/doi.org\/10.37236\/8584","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,5,29]]},"article-number":"P2.35"}}