{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,3]],"date-time":"2023-06-03T05:10:29Z","timestamp":1685769029678},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2010,9,3]],"date-time":"2010-09-03T00:00:00Z","timestamp":1283472000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>Let<jats:italic>k<\/jats:italic>\u2265 1 be an integer, and let<jats:italic>H<\/jats:italic>be a graph with no isolated vertices embedded in the projective plane, such that every homotopically non-trivial closed curve intersects<jats:italic>H<\/jats:italic>at least<jats:italic>k<\/jats:italic>times, and the deletion and contraction of any edge in this embedding results in an embedding that no longer has this property. Let<jats:italic>G<\/jats:italic>be the planar double cover of<jats:italic>H<\/jats:italic>obtained by lifting<jats:italic>G<\/jats:italic>into the universal covering space of the projective plane, the sphere. We prove that<jats:italic>G<\/jats:italic>is minor-minimal of branch-width 2<jats:italic>k<\/jats:italic>. We also exhibit examples of minor-minimal planar graphs of branch-width 6 that do not arise in this way.<\/jats:p>","DOI":"10.1017\/s0963548310000283","type":"journal-article","created":{"date-parts":[[2010,9,3]],"date-time":"2010-09-03T08:44:26Z","timestamp":1283503466000},"page":"73-82","source":"Crossref","is-referenced-by-count":0,"title":["Minor-Minimal Planar Graphs of Even Branch-Width"],"prefix":"10.1017","volume":"20","author":[{"given":"TORSTEN","family":"INKMANN","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ROBIN","family":"THOMAS","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2010,9,3]]},"reference":[{"key":"S0963548310000283_ref14","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/147\/01182"},{"key":"S0963548310000283_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(91)90036-J"},{"key":"S0963548310000283_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(90)90120-O"},{"key":"S0963548310000283_ref8","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0118(199706)25:2<153::AID-JGT7>3.0.CO;2-L"},{"key":"S0963548310000283_ref7","doi-asserted-by":"crossref","DOI":"10.56021\/9780801866890","volume-title":"Graphs on Surfaces","author":"Mohar","year":"2001"},{"key":"S0963548310000283_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(81)90042-3"},{"key":"S0963548310000283_ref5","unstructured":"[5] Johnson E. and Robertson N. Private communication."},{"key":"S0963548310000283_ref2","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.1999.1011"},{"key":"S0963548310000283_ref12","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1994.1046"},{"key":"S0963548310000283_ref1","first-page":"63","volume-title":"Applied Geometry and Discrete Mathematics","author":"Barnette","year":"1991"},{"key":"S0963548310000283_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(92)90038-Y"},{"key":"S0963548310000283_ref4","unstructured":"[4] Inkmann T. (2008) Tree-based decompositions of graphs on surfaces and applications to the traveling salesman problem. PhD dissertation, Georgia Institute of Technology."},{"key":"S0963548310000283_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01215352"},{"key":"S0963548310000283_ref3","unstructured":"[3] Dharmatilake J. (1994) Binary matroids with branch-width three. PhD thesis, Ohio State University."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548310000283","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,3]],"date-time":"2023-06-03T04:30:00Z","timestamp":1685766600000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548310000283\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,9,3]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["S0963548310000283"],"URL":"https:\/\/doi.org\/10.1017\/s0963548310000283","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,9,3]]}}}