{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T05:53:27Z","timestamp":1698213207676},"reference-count":11,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":5397,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1992,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>Q<\/jats:italic>(<jats:italic>n<\/jats:italic>) be the <jats:italic>n<\/jats:italic>\u2010dimensional hypercube, and <jats:italic>X<\/jats:italic>, a set of points in <jats:italic>Q<\/jats:italic>(<jats:italic>n<\/jats:italic>). The Steiner problem for the hypercube is to find the smallest possible number <jats:italic>L<\/jats:italic>(<jats:italic>n,X<\/jats:italic>) of edges in any subtree of <jats:italic>Q<\/jats:italic>(<jats:italic>n<\/jats:italic>) that spans <jats:italic>X<\/jats:italic>. We obtain the following results:\n<jats:list list-type=\"explicit-label\">\n<jats:list-item><jats:p>An exact formula for <jats:italic>L<\/jats:italic>(<jats:italic>n,X<\/jats:italic>), when |<jats:italic>X<\/jats:italic>| \u2264 5.<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>The bound <jats:italic>L<\/jats:italic>(<jats:italic>n,X<\/jats:italic>) \u2264 (<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup><jats:sub><jats:italic>k<\/jats:italic>+1<\/jats:sub>) + (2 + <jats:italic>o<\/jats:italic>(1)) ([log (<jats:italic>k<\/jats:italic>)]\/<jats:italic>k<\/jats:italic>)(<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>) as <jats:italic>k<\/jats:italic> \u2192 \u221e, when <jats:italic>X<\/jats:italic> is the set of all points in <jats:italic>Q<\/jats:italic>(<jats:italic>n<\/jats:italic>) of a given weight <jats:italic>k<\/jats:italic> + 1, provided (<jats:italic>k<\/jats:italic><jats:sup>2<\/jats:sup>\/[log (<jats:italic>k<\/jats:italic>)])<jats:sup>1 + 1\/<jats:italic>k<\/jats:italic><\/jats:sup> \u2264 <jats:italic>n<\/jats:italic>.<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>NP\u2010completeness of deciding <jats:italic>L<\/jats:italic>(<jats:italic>n,X<\/jats:italic>) even when every point of <jats:italic>X<\/jats:italic> has weight at most 2.<\/jats:p><\/jats:list-item>\n<\/jats:list><\/jats:p>","DOI":"10.1002\/net.3230220102","type":"journal-article","created":{"date-parts":[[2007,5,12]],"date-time":"2007-05-12T11:49:35Z","timestamp":1178970575000},"page":"1-19","source":"Crossref","is-referenced-by-count":4,"title":["The steiner problem in the hypercube"],"prefix":"10.1002","volume":"22","author":[{"given":"Zevi","family":"Miller","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Manley","family":"Perkel","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":"10.1002\/net.3230070104"},{"key":"e_1_2_1_3_2","unstructured":"D.deCaen On Turan's hypergraph problem. Ph.D. Thesis University of Toronto (1982)."},{"key":"e_1_2_1_4_2","first-page":"277","article-title":"On constructive upper bounds for the Turan numbers T(n,2r+1,2r)","volume":"65","author":"deCaen D.","year":"1988","journal-title":"Congress. Numer."},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0196-8858(82)80004-3"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02582949"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1137\/0132071"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1137\/0130013"},{"key":"e_1_2_1_9_2","unstructured":"F. K.HwangandD.Richards Steiner tree problems. Networks in press."},{"key":"e_1_2_1_10_2","first-page":"436","article-title":"An extremal problem in graph theory (in Hungarian)","volume":"48","author":"Turan P.","year":"1941","journal-title":"Mat. Fiz. Lapok"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230170203"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5564(86)90161-6"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230220102","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230220102","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T13:42:35Z","timestamp":1698154955000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230220102"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,1]]},"references-count":11,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1992,1]]}},"alternative-id":["10.1002\/net.3230220102"],"URL":"https:\/\/doi.org\/10.1002\/net.3230220102","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,1]]}}}