{"id":{"repo_id":"lsu-thes","oai_identifier":"oai:repository.lsu.edu:gradschool_dissertations-1897"},"canonical_url":"https://search.dev.ndltd.org/etd/lsu-thes/oai:repository.lsu.edu:gradschool_dissertations-1897","repository":{"repo_id":"lsu-thes","name":"Lousiana State University","base_url":"https://repository.lsu.edu/do/oai/"},"display":{"title":"The structure of 4-separations in 4-connected matroids","abstract":"Oxley, Semple and Whittle described a tree decomposition for a 3-connected matroid M that displays, up to a natural equivalence, all non-trivial 3-separations of M. Crossing 3-separations gave rise to fundamental structures known as flowers. In this dissertation, we define generalized flower structure called a k-flower, with no assumptions on the connectivity of M. We completely classify k-flowers in terms of the local connectivity between pairs of petals. Specializing to the case of 4-connected matroids, we give a new notion of equivalence of 4-separations that we show will be needed to describe a tree decomposition for 4-connected matroids. Finally, we characterize all internally 4-connected binary matroids M with the property that the ground set of M can be cyclically ordered so that any consecutive collection of elements in this cyclic ordering is 4-separating. We prove that in this case either M is a matroid on at most seven elements or, up to duality, M is isomorphic to the polygon matroid of a cubic or quartic planar ladder, the polygon matroid of a cubic or quartic Möbius ladder, a particular single-element extension of a wheel, or a particular single-element extension of the bond matroid of a cubic ladder.","abstract_html":"Oxley, Semple and Whittle described a tree decomposition for a 3-connected matroid M that displays, up to a natural equivalence, all non-trivial 3-separations of M. Crossing 3-separations gave rise to fundamental structures known as flowers. In this dissertation, we define generalized flower structure called a k-flower, with no assumptions on the connectivity of M. We completely classify k-flowers in terms of the local connectivity between pairs of petals. Specializing to the case of 4-connected matroids, we give a new notion of equivalence of 4-separations that we show will be needed to describe a tree decomposition for 4-connected matroids. Finally, we characterize all internally 4-connected binary matroids M with the property that the ground set of M can be cyclically ordered so that any consecutive collection of elements in this cyclic ordering is 4-separating. We prove that in this case either M is a matroid on at most seven elements or, up to duality, M is isomorphic to the polygon matroid of a cubic or quartic planar ladder, the polygon matroid of a cubic or quartic Möbius ladder, a particular single-element extension of a wheel, or a particular single-element extension of the bond matroid of a cubic ladder.","abstract_has_math":false,"creators":["Aikin, Jeremy M."],"institution":"Mathematics","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Applied Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-01-01T08:00:00Z","date_published":"2009-01-01T08:00:00Z","updated_at":"2026-07-24T02:58:25Z","subjects":["matroid","connectivity","separation","flowers","local connectivity","pods","2-equivalence"],"languages":[],"rights":["unrestricted","Release the entire work immediately for access worldwide."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-07082009-102124","https://repository.lsu.edu/gradschool_dissertations/898"],"render_values":[{"text":"etd-07082009-102124","href":null,"code":true},{"text":"https://repository.lsu.edu/gradschool_dissertations/898","href":"https://repository.lsu.edu/gradschool_dissertations/898","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.31390/gradschool_dissertations.898","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Aikin, Jeremy M."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2009-06-12"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-05-12T23:10:21Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Applied Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["matroid","connectivity","separation","flowers","local connectivity","pods","2-equivalence"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","Release the entire work immediately for access worldwide."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-07082009-102124","10.31390/gradschool_dissertations.898","https://repository.lsu.edu/gradschool_dissertations/898"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Oxley, Semple and Whittle described a tree decomposition for a 3-connected matroid M that displays, up to a natural equivalence, all non-trivial 3-separations of M. Crossing 3-separations gave rise to fundamental structures known as flowers. In this dissertation, we define generalized flower structure called a k-flower, with no assumptions on the connectivity of M. We completely classify k-flowers in terms of the local connectivity between pairs of petals. Specializing to the case of 4-connected matroids, we give a new notion of equivalence of 4-separations that we show will be needed to describe a tree decomposition for 4-connected matroids. Finally, we characterize all internally 4-connected binary matroids M with the property that the ground set of M can be cyclically ordered so that any consecutive collection of elements in this cyclic ordering is 4-separating. We prove that in this case either M is a matroid on at most seven elements or, up to duality, M is isomorphic to the polygon matroid of a cubic or quartic planar ladder, the polygon matroid of a cubic or quartic Möbius ladder, a particular single-element extension of a wheel, or a particular single-element extension of the bond matroid of a cubic ladder."]},{"key":"dc:title","label":"Title","values":["The structure of 4-separations in 4-connected matroids"]}]}],"canonical_facts":{"dc:creator":["Aikin, Jeremy M."],"dc:date":["2009-06-12"],"dc:date.available":["2022-05-12T23:10:21Z"],"dc:description.abstract":["Oxley, Semple and Whittle described a tree decomposition for a 3-connected matroid M that displays, up to a natural equivalence, all non-trivial 3-separations of M. Crossing 3-separations gave rise to fundamental structures known as flowers. In this dissertation, we define generalized flower structure called a k-flower, with no assumptions on the connectivity of M. We completely classify k-flowers in terms of the local connectivity between pairs of petals. Specializing to the case of 4-connected matroids, we give a new notion of equivalence of 4-separations that we show will be needed to describe a tree decomposition for 4-connected matroids. Finally, we characterize all internally 4-connected binary matroids M with the property that the ground set of M can be cyclically ordered so that any consecutive collection of elements in this cyclic ordering is 4-separating. We prove that in this case either M is a matroid on at most seven elements or, up to duality, M is isomorphic to the polygon matroid of a cubic or quartic planar ladder, the polygon matroid of a cubic or quartic Möbius ladder, a particular single-element extension of a wheel, or a particular single-element extension of the bond matroid of a cubic ladder."],"dc:identifier":["etd-07082009-102124","10.31390/gradschool_dissertations.898","https://repository.lsu.edu/gradschool_dissertations/898"],"dc:rights":["unrestricted","Release the entire work immediately for access worldwide."],"dc:subject":["matroid","connectivity","separation","flowers","local connectivity","pods","2-equivalence"],"dc:title":["The structure of 4-separations in 4-connected matroids"],"thesis:degree_discipline":["Applied Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"],"thesis:institution_name":["Mathematics"]},"updated_at":"2026-07-24T02:58:25Z"}