{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T09:44:39Z","timestamp":1698227079311},"reference-count":21,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2006,10,6]],"date-time":"2006-10-06T00:00:00Z","timestamp":1160092800000},"content-version":"vor","delay-in-days":4449,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1994,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Given a digraph <jats:italic>D<\/jats:italic> on vertices <jats:italic>v<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>v<\/jats:italic><jats:sub>2<\/jats:sub>,  \u20db, <jats:italic>v<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>, we can associate a bipartite graph <jats:italic>B(D)<\/jats:italic> on vertices <jats:italic>s<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>s<\/jats:italic><jats:sub>2<\/jats:sub>,  \u20db, <jats:italic>s<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>, <jats:italic>t<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>t<\/jats:italic><jats:sub>2<\/jats:sub>,  \u20db, <jats:italic>t<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>, where <jats:italic>s<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub><jats:italic>t<\/jats:italic><jats:sub><jats:italic>j<\/jats:italic><\/jats:sub> is an edge of <jats:italic>B(D)<\/jats:italic> if (<jats:italic>v<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, <jats:italic>v<\/jats:italic><jats:sub><jats:italic>j<\/jats:italic><\/jats:sub>) is an arc in <jats:italic>D<\/jats:italic>. Let <jats:italic>O<\/jats:italic><jats:sub><jats:italic>G<\/jats:italic><\/jats:sub> denote the set of all orientations on the (undirected) graph <jats:italic>G<\/jats:italic>. In this paper we will discuss properties of the set <jats:italic>S<\/jats:italic>(<jats:italic>G<\/jats:italic>) := {\u03b2<jats:sub>1<\/jats:sub> (<jats:italic>B<\/jats:italic>(<jats:italic>D<\/jats:italic>))) | <jats:italic>D<\/jats:italic> \u2245 <jats:italic>O<\/jats:italic><jats:sub><jats:italic>G<\/jats:italic><\/jats:sub>}, where \u03b2<jats:sub>1<\/jats:sub> is the edge independence number.<\/jats:p><jats:p>In the first section we present some background and related concepts. We show that sets of the form <jats:italic>S<\/jats:italic>(<jats:italic>G<\/jats:italic>) are convex and that max <jats:italic>S<\/jats:italic>(<jats:italic>G<\/jats:italic>) \u2266 2 min <jats:italic>S<\/jats:italic>(<jats:italic>G<\/jats:italic>). Furthermore, this completely characterizes such sets.<\/jats:p><jats:p>In the second section we discuss some bounds on elements of <jats:italic>S<\/jats:italic>(<jats:italic>G<\/jats:italic>) in terms of more familiar graphical parameters.<\/jats:p><jats:p>The third section deals with extremal problems. We discuss bounds on elements of <jats:italic>S<\/jats:italic>(<jats:italic>G<\/jats:italic>) if the order and size of <jats:italic>G<\/jats:italic> are known, particularly when <jats:italic>G<\/jats:italic> is bipartite. In this section we exhibit a relation between max <jats:italic>S<\/jats:italic>(<jats:italic>G<\/jats:italic>) and the concept of graphical closure.<\/jats:p><jats:p>In the fourth and final section we discuss the computational complexity of computing min <jats:italic>S<\/jats:italic>(<jats:italic>G<\/jats:italic>) and max <jats:italic>S<\/jats:italic>(<jats:italic>G<\/jats:italic>). We show that the first problem is NP\u2010complete and that the latter is polynomial.<\/jats:p>","DOI":"10.1002\/jgt.3190180509","type":"journal-article","created":{"date-parts":[[2007,6,7]],"date-time":"2007-06-07T18:25:06Z","timestamp":1181240706000},"page":"515-533","source":"Crossref","is-referenced-by-count":0,"title":["Independent edges in bipartite graphs obtained from orientations of graphs"],"prefix":"10.1002","volume":"18","author":[{"given":"J. G.","family":"Gimbel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"K. 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