{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,27]],"date-time":"2024-03-27T19:23:22Z","timestamp":1711567402899},"reference-count":11,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2007,3,29]],"date-time":"2007-03-29T00:00:00Z","timestamp":1175126400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[2007,8]]},"abstract":"Abstract<\/jats:title>A well\u2010known formula of Tutte and Berge expresses the size of a maximum matching in a graph G<\/jats:italic> in terms of what is usually called the deficiency of G<\/jats:italic>. A subset X<\/jats:italic> of V<\/jats:italic>(G<\/jats:italic>) for which this deficiency is attained is called a Tutte set of G<\/jats:italic>. While much is known about maximum matchings, less is known about the structure of Tutte sets. In this article, we study the structural aspects of maximal Tutte sets in a graph G<\/jats:italic>. Towards this end, we introduce a related graph D<\/jats:italic>(G<\/jats:italic>). We first show that the maximal Tutte sets in G<\/jats:italic> are precisely the maximal independent sets in its D<\/jats:italic>\u2010graph D<\/jats:italic>(G<\/jats:italic>), and then continue with the study of D<\/jats:italic>\u2010graphs in their own right, and of iterated D<\/jats:italic>\u2010graphs. We show that G<\/jats:italic> is isomorphic to a spanning subgraph of D<\/jats:italic>(G<\/jats:italic>), and characterize the graphs for which G<\/jats:italic>\u2245D<\/jats:italic>(G<\/jats:italic>) and for which D<\/jats:italic>(G<\/jats:italic>)\u2245D<\/jats:italic>2<\/jats:sup>(G<\/jats:italic>). Surprisingly, it turns out that for every graph G<\/jats:italic> with a perfect matching, D<\/jats:italic>3<\/jats:sup>(G<\/jats:italic>)\u2245D<\/jats:italic>2<\/jats:sup>(G<\/jats:italic>). Finally, we characterize bipartite D<\/jats:italic>\u2010graphs and comment on the problem of characterizing D<\/jats:italic>\u2010graphs in general. \u00a9 2007 Wiley Periodicals, Inc. J Graph Theory 55: 343\u2013358, 2007<\/jats:p>","DOI":"10.1002\/jgt.20243","type":"journal-article","created":{"date-parts":[[2007,3,29]],"date-time":"2007-03-29T14:46:41Z","timestamp":1175179601000},"page":"343-358","source":"Crossref","is-referenced-by-count":4,"title":["Tutte sets in graphs I: Maximal tutte sets and D\u2010graphs"],"prefix":"10.1002","volume":"55","author":[{"given":"D.","family":"Bauer","sequence":"first","affiliation":[]},{"given":"H. J.","family":"Broersma","sequence":"additional","affiliation":[]},{"given":"A.","family":"Morgana","sequence":"additional","affiliation":[]},{"given":"E.","family":"Schmeichel","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2007,3,29]]},"reference":[{"key":"e_1_2_1_2_2","first-page":"49","article-title":"Isolating the families of soliton graphs","volume":"13","author":"Bartha M.","year":"2002","journal-title":"Pure Mathematics and Applications"},{"key":"e_1_2_1_3_2","unstructured":"M.BarthaandM.Kresz Splitters and barriers in open graphs having a perfect internal matching Preprint."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(02)00048-8"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2004.05.016"},{"key":"e_1_2_1_6_2","article-title":"Tutte sets in graphs II: the complexity of finding maximum Tutte sets","author":"Bauer D.","journal-title":"Discrete Applied Mathematics"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.43.9.842"},{"key":"e_1_2_1_8_2","first-page":"258","article-title":"Sur le couplage maximum d'un graphe","volume":"247","author":"Berge C.","year":"1958","journal-title":"C R Acad Sci (Paris)"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.6028\/jres.069B.013"},{"key":"e_1_2_1_10_2","volume-title":"Matching Theory, Ann Discrete Math Vol 29","author":"Lov\u00e1sz L.","year":"1986"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-22.2.107"},{"key":"e_1_2_1_12_2","volume-title":"Introduction to Graph Theory, second edition","author":"West D. B.","year":"2001"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.20243","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.20243","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T10:53:13Z","timestamp":1697626393000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.20243"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,3,29]]},"references-count":11,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2007,8]]}},"alternative-id":["10.1002\/jgt.20243"],"URL":"https:\/\/doi.org\/10.1002\/jgt.20243","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,3,29]]}}}