{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,22]],"date-time":"2023-10-22T06:41:01Z","timestamp":1697956861929},"reference-count":5,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2007,3,21]],"date-time":"2007-03-21T00:00:00Z","timestamp":1174435200000},"content-version":"vor","delay-in-days":7749,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Systems &amp;amp; Computers in Japan"],"published-print":{"date-parts":[[1986,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Given a graph and a positive integer <jats:italic>k<\/jats:italic>, in the <jats:italic>k<\/jats:italic>\u2010node\u2010connectivity augmentation problem (<jats:italic>k<\/jats:italic>\u2010NCAP), a set of edges is determined to be added to convert the graph into a <jats:italic>k<\/jats:italic>\u2010node\u2010connected graph with the minimum sum of the added edge\u2010weights. It is known that 1\u2010NCAP, for the directed acyclic graph where the weight is restricted to 1 and 2, is NP\u2010complete. It is also known that when the weight of the edges are all equal, 1\u2010NCAP for any directed graph can be solved in <jats:italic>O<\/jats:italic>([<jats:italic>V<\/jats:italic>] + [<jats:italic>E<\/jats:italic>]) time, and when the graph is restricted to the directed binary tree, <jats:italic>k<\/jats:italic>\u2010NCAP (<jats:italic>k<\/jats:italic> \u2265 2) is solved in <jats:italic>O<\/jats:italic>(<jats:italic>k<\/jats:italic> |y|) time. In the preceding, |V and |E are the number of nodes and the number of edges of the given graph, respectively. This paper discusses <jats:italic>k<\/jats:italic>\u2010NCAP (<jats:italic>k<\/jats:italic> \u2265 2) for the directed graph, where all edge weights are equal. A solution for <jats:italic>k<\/jats:italic>\u2010NCAP is shown for the family of graphs properly containing the directed ternary trees. It is shown that the problem can be solved in (<jats:italic>k<\/jats:italic> \u2265 3) time <jats:italic>O<\/jats:italic>(<jats:italic>k<\/jats:italic> |V) for the directed ternary tree, which is the optimal\u2010time algorithm within a constant factor. In this paper, a new family of regular and <jats:italic>k<\/jats:italic>\u2010node\u2010connected graphs called a <jats:italic>k<\/jats:italic>\u2010bouquet is defined. The solution for <jats:italic>k<\/jats:italic>\u2010NCAP is obtained by constructing a <jats:italic>k<\/jats:italic>\u2010bouquet by adding edges to the given graph.<\/jats:p>","DOI":"10.1002\/scj.4690170307","type":"journal-article","created":{"date-parts":[[2007,7,7]],"date-time":"2007-07-07T11:23:19Z","timestamp":1183807399000},"page":"56-65","source":"Crossref","is-referenced-by-count":0,"title":["Optimal\u2010Time Algorithm for the k\u2010Node\u2010Connectivity Augmentation Problem for Ternary Trees"],"prefix":"10.1002","volume":"17","author":[{"given":"Toshimitsu","family":"Masuzawa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ken'ichi","family":"Hagihara","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nobuki","family":"Tokura","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Koichi","family":"Wada","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2007,3,21]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Data Structures and Algorithms","author":"Aho A. V.","year":"1983"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1137\/0205044"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1137\/0210019"},{"key":"e_1_2_1_5_2","first-page":"1","article-title":"The k\u2010node\u2010connectivity augmentation problem for directed binary trees","volume":"67","author":"Masuzawa T.","year":"1984","journal-title":"Trans. I.E.C.E., Japan"},{"key":"e_1_2_1_6_2","first-page":"2","article-title":"The graph augmentation problems with respect to node\u2010connectivity (I)","volume":"84","author":"Masuzawa T.","year":"1984","journal-title":"Tech. Rep., I.E.C.E., Japan"}],"container-title":["Systems and Computers in Japan"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fscj.4690170307","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/scj.4690170307","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,22]],"date-time":"2023-10-22T00:29:07Z","timestamp":1697934547000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/scj.4690170307"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,1]]},"references-count":5,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1986,1]]}},"alternative-id":["10.1002\/scj.4690170307"],"URL":"https:\/\/doi.org\/10.1002\/scj.4690170307","archive":["Portico"],"relation":{},"ISSN":["0882-1666","1520-684X"],"issn-type":[{"value":"0882-1666","type":"print"},{"value":"1520-684X","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,1]]}}}