{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T03:09:06Z","timestamp":1761620946635},"reference-count":13,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2006,11,6]],"date-time":"2006-11-06T00:00:00Z","timestamp":1162771200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[2007,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Boesch et al. conjectured that for any <jats:italic>n<\/jats:italic> and <jats:italic>m<\/jats:italic> there exists a uniformly optimal (<jats:italic>n<\/jats:italic>,<jats:italic>m<\/jats:italic>)\u2013graph <jats:italic>G<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>,<jats:italic>m<\/jats:italic><\/jats:sub> for all terminal reliability, that is, the all\u2010terminal reliability of <jats:italic>G<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>,<jats:italic>m<\/jats:italic><\/jats:sub> is at least as large as the all\u2010terminal reliability of any other graph <jats:italic>G<\/jats:italic> with <jats:italic>n<\/jats:italic> vertices and <jats:italic>m<\/jats:italic> edges, no matter what the probability of an edge being operational is. Although there are counterexamples known when one restricts attention to simple graphs, the conjecture remains open when one allows parallel edges. We consider the analogous problem for strongly connected reliability, that is, the probability that a digraph contains a spanning strongly connected subdigraph, given that each vertex is operational, but arcs are independently operational with probability <jats:italic>p<\/jats:italic>. We show that there do indeed exist uniformly optimal digraphs for strongly connected (<jats:italic>n<\/jats:italic>,<jats:italic>m<\/jats:italic>)\u2013digraphs. We also show that if one restricts attention to simple digraphs (without parallel arcs) then such uniformly optimal digraphs need not exist. \u00a9 2006 Wiley Periodicals, Inc. 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