{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T09:42:54Z","timestamp":1698313374963},"reference-count":30,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":4850,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1993,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Emulations are a special kind of structure\u2010preserving mappings between processor interconnection networks (graphs). In this paper, the notion of emulation is generalized to the notion of <jats:italic>d<\/jats:italic>\u2010emulation for any <jats:italic>d<\/jats:italic> \u2267 1. Several problems concerning the complexity of (uniform) <jats:italic>d<\/jats:italic>\u2010emulations are studied. It is shown that the problem of deciding for arbitrary graphs <jats:italic>H<\/jats:italic> and <jats:italic>G<\/jats:italic> whether <jats:italic>H<\/jats:italic> can be uniformly <jats:italic>d<\/jats:italic>\u2010emulated on <jats:italic>G<\/jats:italic> is NP\u2010complete for every fixed <jats:italic>d<\/jats:italic> \u03f5 N. Also, it is shown that the problem of deciding for arbitrary rings <jats:italic>R<\/jats:italic> and planar graphs <jats:italic>G<\/jats:italic> whether <jats:italic>R<\/jats:italic> can be uniformly 2\u2010emulated on <jats:italic>G<\/jats:italic> is NP\u2010complete. Further, a constructive proof is given of the fact that there always exists a uniform 3\u2010emulation of a ring <jats:italic>R<\/jats:italic> on a graph <jats:italic>G<\/jats:italic> = <jats:italic>(V, E)<\/jats:italic> if |<jats:italic>V<\/jats:italic><jats:sub><jats:italic>R<\/jats:italic><\/jats:sub>y| = <jats:italic>k<\/jats:italic>. |<jats:italic>V<\/jats:italic><jats:sub><jats:italic>G<\/jats:italic><\/jats:sub>| and <jats:italic>k<\/jats:italic> \u03f5 N. This uniform 3\u2010emulation of <jats:italic>R<\/jats:italic> on <jats:italic>G<\/jats:italic> is used as the basis for the Distributed Virtual Ring Construction Algorithm that only uses 2|<jats:italic>E<\/jats:italic>| messages of <jats:italic>O<\/jats:italic>(1) bits and has time complexity \u2266 2(|<jats:italic>V<\/jats:italic>| \u2212 1). A traversal of the constructed virtual ring costs 2(|<jats:italic>V<\/jats:italic>| \u2212 1) messages. \u00a9 <jats:italic>1993 John Wiley &amp; Sons, Inc.<\/jats:italic><\/jats:p>","DOI":"10.1002\/net.3230230405","type":"journal-article","created":{"date-parts":[[2007,5,12]],"date-time":"2007-05-12T14:21:36Z","timestamp":1178979696000},"page":"237-248","source":"Crossref","is-referenced-by-count":1,"title":["Uniform <i>d<\/i>\u2010emulations of rings, with an application to distributed virtual ring construction"],"prefix":"10.1002","volume":"23","author":[{"given":"Erwin M.","family":"Bakker","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jan","family":"van Leeuwen","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","volume-title":"Distributed Algorithms on Graphs, Proceedings 1st International Workshop on Distributed Algorithms on Graphs, 1985","author":"Andreasson S. 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