{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T09:22:08Z","timestamp":1698225728017},"reference-count":14,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2006,10,4]],"date-time":"2006-10-04T00:00:00Z","timestamp":1159920000000},"content-version":"vor","delay-in-days":6882,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1987,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A<jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> is a complete subgraph of size i. A <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2010cover of a graph G(V, E) is a set <jats:bold>C<\/jats:bold> of <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic>\u22121<\/jats:sub><jats:italic>s<\/jats:italic> of <jats:italic>G<\/jats:italic> such that every <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> in <jats:italic>G<\/jats:italic> contains at least one <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic>\u22121<\/jats:sub> in <jats:bold>C<\/jats:bold>. <jats:italic>c<\/jats:italic><jats:sub>i<\/jats:sub><jats:italic>(G)<\/jats:italic> is the cardinality of a smallest <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2010cover of <jats:italic>G<\/jats:italic>. A <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2010packing of <jats:italic>G<\/jats:italic> is a set of <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub><jats:italic>s<\/jats:italic> such that no two <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub><jats:italic>s<\/jats:italic> have <jats:italic>i<\/jats:italic> \u2212 1 nodes in common. <jats:italic>p<jats:sub>i<\/jats:sub>(G)<\/jats:italic> is the cardinality of a largest <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2010packing of <jats:italic>G<\/jats:italic>. Let <jats:bold>F<\/jats:bold><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub><jats:italic>(G)<\/jats:italic> denote the set of <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>s in <jats:italic>G<\/jats:italic> and define <jats:italic>c<jats:sub>i<\/jats:sub>(F)<\/jats:italic> and <jats:italic>p<jats:sub>i<\/jats:sub>(F)<\/jats:italic> analogously for <jats:italic>F<\/jats:italic> \u2286 <jats:bold>F<\/jats:bold><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>(G). <jats:italic>G<\/jats:italic> is <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2010perfect if \u2200<jats:italic>F<\/jats:italic> \u2286 <jats:bold>F<\/jats:bold><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>(G), c<jats:sub>i<\/jats:sub>(F) = p<jats:sub>i<\/jats:sub><jats:italic>(F)<\/jats:italic>. The <jats:italic>K<\/jats:italic><jats:sub>2<\/jats:sub>\u2010perfect graphs are precisely the bipartite graphs. We present a characterization of <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2010perfect graphs that is similar to the Strong Perfect Graph Conjecture, and explore the relationships between <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2010perfect graphs and normal hypergraphs. Furthermore, if <jats:italic>iA<\/jats:italic> denotes the 0 \u2212 1 matrix of <jats:italic>G<\/jats:italic> where the rows are the elements of <jats:bold>F<\/jats:bold><jats:sub><jats:italic>i<\/jats:italic>\u22121<\/jats:sub>(<jats:italic>G<\/jats:italic>) that belong to at least one <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> and the columns are the elements of <jats:bold>F<\/jats:bold><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub><jats:italic>(G)<\/jats:italic>, then we show that <jats:italic>iA<\/jats:italic> is perfect iff <jats:italic>G<\/jats:italic> is a <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2010perfect graph. We also characterize the <jats:italic>K<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2010perfect graphs for which <jats:italic>iA<\/jats:italic>is balanced.<\/jats:p>","DOI":"10.1002\/jgt.3190110415","type":"journal-article","created":{"date-parts":[[2007,6,9]],"date-time":"2007-06-09T05:08:32Z","timestamp":1181365712000},"page":"569-584","source":"Crossref","is-referenced-by-count":8,"title":["<i>K<\/i><sub><i>i<\/i><\/sub>\u2010covers. II. <i>K<\/i><sub><i>i<\/i><\/sub>\u2010perfect graphs"],"prefix":"10.1002","volume":"11","author":[{"given":"Michele","family":"Conforti","sequence":"first","affiliation":[]},{"given":"Derek Gordon","family":"Corneil","sequence":"additional","affiliation":[]},{"given":"Ali Ridha","family":"Mahjoub","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,4]]},"reference":[{"key":"e_1_2_1_2_2","first-page":"155","volume-title":"Graph Theory and Theoretical Physics","author":"Berge C.","year":"1967"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01584535"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(79)90067-4"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(75)90041-6"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(86)90156-1"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(85)90027-7"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(81)90013-5"},{"key":"e_1_2_1_9_2","first-page":"155","article-title":"Analogues of the Shannon capacity of a graph","volume":"12","author":"Hell P.","year":"1982","journal-title":"Ann. 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