{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T22:25:29Z","timestamp":1757543129715},"reference-count":2,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":7437,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1986,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For a directed graph <jats:italic>G<\/jats:italic>, let <jats:italic>d<\/jats:italic><jats:sup>+<\/jats:sup>(\u03bd) and <jats:italic>d<\/jats:italic> (\u03bd) be the outdegree and indegree of vertex \u03bd, respectively. Given a positive integer <jats:italic>k<\/jats:italic>, the outdeficiency and indeficiency are defined by \u03b4<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-stack-1.gif\" xlink:title=\"urn:x-wiley:00283045:media:NET3230160207:tex2gif-stack-1\" \/> (\u03bd) = max (<jats:italic>k<\/jats:italic> \u2212 <jats:italic>d<\/jats:italic><jats:sup>+<\/jats:sup> (\u03bd), 0) and \u03b4<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-stack-2.gif\" xlink:title=\"urn:x-wiley:00283045:media:NET3230160207:tex2gif-stack-2\" \/> (\u03bd) = max(<jats:italic>k<\/jats:italic> \u2212 <jats:italic>d<\/jats:italic><jats:sup>\u2212<\/jats:sup> (\u03bd), 0), respectively. It is evident that in augmenting <jats:italic>G<\/jats:italic> to a <jats:italic>k<\/jats:italic>\u2010edge\u2010connected directed graph, at least \u03b4<jats:sub>k<\/jats:sub>(<jats:italic>G<\/jats:italic>) = max(\u03a3\u03b4<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-stack-3.gif\" xlink:title=\"urn:x-wiley:00283045:media:NET3230160207:tex2gif-stack-3\" \/> (\u03bd), \u03a3\u03b4<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-stack-4.gif\" xlink:title=\"urn:x-wiley:00283045:media:NET3230160207:tex2gif-stack-4\" \/> (\u03bd)) edges are necessary. This paper proves the theorem that if <jats:italic>G<\/jats:italic> is a directed tree (directed graph whose underlying graph is a tree) this number of edges is enough. The proof is made by presenting a construction procedure of polynomial\u2010order complexity.<\/jats:p>","DOI":"10.1002\/net.3230160207","type":"journal-article","created":{"date-parts":[[2007,5,11]],"date-time":"2007-05-11T20:50:09Z","timestamp":1178916609000},"page":"181-197","source":"Crossref","is-referenced-by-count":15,"title":["The minimum augmentation of a directed tree to a <i>k<\/i>\u2010edge\u2010connected directed graph"],"prefix":"10.1002","volume":"16","author":[{"given":"Yoji","family":"Kajitani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shuichi","family":"Ueno","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1137\/0205044"},{"key":"e_1_2_1_3_2","unstructured":"S.Ueno Y.Kajitani andWada The minimum augmentation of a tree to. k\u2010edge\u2010connected graph. (Submitted to Networks 1984)"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230160207","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230160207","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T01:54:42Z","timestamp":1697853282000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230160207"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,6]]},"references-count":2,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1986,6]]}},"alternative-id":["10.1002\/net.3230160207"],"URL":"https:\/\/doi.org\/10.1002\/net.3230160207","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,6]]}}}