{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:38:09Z","timestamp":1787323089400,"version":"build-2736575974"},"reference-count":14,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2026,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>A partitioned matroid [Formula: see text] consists of a matroid [Formula: see text] and a partition [Formula: see text] of its ground set. As such structures arise frequently in structural matroid theory, this paper introduces a general technique for analyzing those special properties of partitioned matroids that depend solely on the values of the connectivities [Formula: see text], the local connectivities [Formula: see text], and the dual local connectivities [Formula: see text]. In particular, we consider those partitioned matroids in which each [Formula: see text] is an independent, coindependent set of clones of cardinality [Formula: see text]. Calling such partitioned matroids clonal-core matroids, we show that special results of the above type for partitioned matroids can be verified in general by proving them just for clonal-core matroids. Aiming at the long-term goal of finding the unavoidable minors of 4-connected matroids, we illustrate this technique by studying 4-paths. These are sequences [Formula: see text] of sets that partition the ground set of a matroid so that the union of any proper initial segment of parts is 4-separating. Viewing the ends [Formula: see text] and [Formula: see text] as fixed, we call such a partition a 4-flexipath if [Formula: see text] is a 4-path for all permutations [Formula: see text] of [Formula: see text]. A straightforward simplification enables us to focus on [Formula: see text]-flexipaths for some [Formula: see text] in [Formula: see text], that is, those 4-flexipaths for which [Formula: see text] and [Formula: see text] for all distinct [Formula: see text] and [Formula: see text]. Our main result for 4-paths is that the only nontrivial case that arises here is when [Formula: see text]. In that case, there are essentially only two possible dual pairs of [Formula: see text]-flexipaths when [Formula: see text].<\/jats:p>","DOI":"10.1137\/24m1663636","type":"journal-article","created":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T08:25:27Z","timestamp":1770884727000},"page":"134-169","source":"Crossref","is-referenced-by-count":0,"title":["Clonal Cores and Flexipaths in Matroids"],"prefix":"10.1137","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1136-418X","authenticated-orcid":true,"given":"Nick","family":"Brettell","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"James","family":"Oxley","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Charles","family":"Semple","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Geoff","family":"Whittle","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,2,12]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2007.05.004"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2011.06.001"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.37236\/12467"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1975-0357163-6"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2014.05.001"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2013.03.002"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1997.1785"},{"key":"ref8","first-page":"745","volume":"5","author":"Ferroni L.","year":"2022","journal-title":"Algbr. 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