{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T05:28:50Z","timestamp":1782365330309,"version":"3.54.5"},"reference-count":9,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":4972,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1993,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A graph or multigraph <jats:italic>G<\/jats:italic>* is <jats:italic>k<\/jats:italic>\u2010edge fault\u2010tolerant with respect to a graph <jats:italic>G<\/jats:italic>, denoted <jats:italic>k<\/jats:italic>\u2010EFT<jats:italic>(G)<\/jats:italic>, if every graph obtained by removing any <jats:italic>k<\/jats:italic> edges from <jats:italic>G<\/jats:italic>* contains <jats:italic>G<\/jats:italic>. We observe that for <jats:italic>k<\/jats:italic> sufficiently large a <jats:italic>k<\/jats:italic>\u2010EFT<jats:italic>(G)<\/jats:italic> graph must be a multigraph, and we present some basic conditions that such multigraphs must meet. We then study the problem of constructing <jats:italic>k<\/jats:italic>\u2010EFT<jats:italic>(G)<\/jats:italic> graphs that are optimal in that they contain the minimum number of edges among all <jats:italic>k<\/jats:italic>\u2010EFT<jats:italic>(G)<\/jats:italic> graphs. Families of optimal <jats:italic>k<\/jats:italic>\u2010EFT<jats:italic>(G)<\/jats:italic> graphs, where <jats:italic>G<\/jats:italic> is the <jats:italic>n<\/jats:italic>\u2010node path or cycle, are presented for all <jats:italic>k<\/jats:italic> and <jats:italic>n<\/jats:italic>. We also give an optimal 1\u2010EFT design for the <jats:italic>n<\/jats:italic>\u2010dimensional hypercube. \u00a9 <jats:italic>1993 by John Wiley &amp; Sons, Inc.<\/jats:italic><\/jats:p>","DOI":"10.1002\/net.3230230207","type":"journal-article","created":{"date-parts":[[2007,5,12]],"date-time":"2007-05-12T14:18:29Z","timestamp":1178979509000},"page":"135-142","source":"Crossref","is-referenced-by-count":61,"title":["Edge fault tolerance in graphs"],"prefix":"10.1002","volume":"23","author":[{"given":"Frank","family":"Harary","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"John P.","family":"Hayes","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"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.3230190608"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.48.7.1142"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.21236\/AD0705364"},{"key":"e_1_2_1_5_2","unstructured":"F.HararyandJ. P.Hayes Node fault tolerance in graphs. In preparation."},{"key":"e_1_2_1_6_2","first-page":"13","article-title":"Isomorphic factorizations. II: Combinatorial designs","volume":"19","author":"Harary F.","year":"1978","journal-title":"Congress. Num."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1109\/TC.1976.1674712"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230120303"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190100111"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190080118"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230230207","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230230207","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T01:20:40Z","timestamp":1698196840000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230230207"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,3]]},"references-count":9,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1993,3]]}},"alternative-id":["10.1002\/net.3230230207"],"URL":"https:\/\/doi.org\/10.1002\/net.3230230207","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,3]]}}}