{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T09:46:50Z","timestamp":1751881610787},"reference-count":5,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,3]],"date-time":"2006-10-03T00:00:00Z","timestamp":1159833600000},"content-version":"vor","delay-in-days":9712,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1980,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The concept of a matroid vertex is introduced. The vertices of a matroid of a 3\u2010connected graph are in one\u2010to\u2010one correspondence with vertices of the graph. Thence directly follows Whitney's theorem that cyclic isomorphism of 3\u2010connected graphs implies isomorphism. The concept of a vertex of a matroid leads to an equally simple proof of Whitney's theorem on the unique embedding of a 3\u2010connected planar graph in the sphere. It also leads to a number of new facts about 3\u2010connected graphs. Thus, consideration of a vertex in a matroid that is the dual of the matroid of a graph leads to a natural concept of a nonseparating cycle of a graph. Whitney's theorem on cyclic isomorphism can be strengthened (even if the nonseparating cycles of a graph are considered, the theorem is found to work) and a new criterion for planarity of 3\u2010connected graphs is obtained (in terms of nonseparating cycles).<\/jats:p>","DOI":"10.1002\/jgt.3190040103","type":"journal-article","created":{"date-parts":[[2007,5,29]],"date-time":"2007-05-29T06:19:54Z","timestamp":1180419594000},"page":"13-19","source":"Crossref","is-referenced-by-count":8,"title":["Concept of a vertex in a matroid and 3\u2010connected graphs"],"prefix":"10.1002","volume":"4","author":[{"given":"A. K.","family":"Kelmans","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,3]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.21236\/AD0705364"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0060114"},{"key":"e_1_2_1_4_2","volume-title":"Matroid Theory","author":"Welsh D.","year":"1976"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.2307\/2371086"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.2307\/2371125"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190040103","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190040103","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,7]],"date-time":"2023-10-07T15:25:50Z","timestamp":1696692350000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190040103"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,3]]},"references-count":5,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1980,3]]}},"alternative-id":["10.1002\/jgt.3190040103"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190040103","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1980,3]]}}}