{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T04:48:07Z","timestamp":1770958087406,"version":"3.50.1"},"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>The breadth of a tangle $\\mathcal{T}$ in a matroid is the size of the largest spanning uniform submatroid of the tangle matroid of $\\mathcal{T}$. The matroid $M$ is weakly $4$-connected if it is 3-connected and whenever $(X,Y)$ is a partition of $E(M)$ with $|X|,|Y|&gt;4$, then $\\lambda(X)\\geq 3$. We prove that if $\\mathcal{T}$ is a tangle of order $k\\geq 4$ and breadth $l$ in a matroid $M$, then $M$ has a weakly 4-connected minor $N$ with a tangle $\\mathcal{T}_N$ of order $k$, breadth $l$ and has the property that $\\mathcal{T}$ is the tangle in $M$ induced by $\\mathcal{T}_N$.\r\nA set $Z$ of elements of a matroid $M$ is $4$-connected if $\\lambda(A)\\geq\\min\\{|A\\cap Z|,|Z-A|,3\\}$ for all $A\\subseteq E(M)$. As a corollary of our theorems on tangles we prove that if $M$ contains an $n$-element $4$-connected set where $n\\geq 7$, then $M$ has a weakly $4$-connected minor that contains an $n$-element $4$-connected set.<\/jats:p>","DOI":"10.37236\/12467","type":"journal-article","created":{"date-parts":[[2025,4,11]],"date-time":"2025-04-11T14:03:08Z","timestamp":1744380188000},"source":"Crossref","is-referenced-by-count":1,"title":["What is a $4$-Connected Matroid?"],"prefix":"10.37236","volume":"32","author":[{"given":"Nick","family":"Brettell","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Susan","family":"Jowett","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"James","family":"Oxley","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Charles","family":"Semple","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Geoff","family":"Whittle","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,4,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p1\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,11]],"date-time":"2025-04-11T14:03:08Z","timestamp":1744380188000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i2p1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,11]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,4,11]]}},"URL":"https:\/\/doi.org\/10.37236\/12467","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,11]]},"article-number":"P2.1"}}