Unformatted text preview: them all oﬀ, but the trouble was that whatever pairing he tried there were always some couples that didn’t like each other. So he summoned Merlin, the Wizard and ordered him to ﬁnd a pairing in which every pair was willing to marry. Merlin had supernatural powers (or maybe just knew a little about optimization. ..who knows?) and he saw at once that no such pairing was possible. How could he convince the king (who would surely order Merlin to be beheaded if the wizard started babbling about bipartite graphs, ﬂows and cuts)? Below is a simple example with 5 ladies and 5 knights. Convince yourself that there is no perfect matching in this case. How could you convince King Arthur about this? (Lines join nodes corresponding to ladies and knights who are willing to be paired with one another.) Ladies Knights...
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 Fall '05
 TROTTER
 Bipartite graph, Dulmage–Mendelsohn decomposition, bipartite graphs

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