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Unformatted text preview: e} and T = {t} whose capacity is Cat + Cet = 5 + 4 = 9. (b) With arcs ab, ee, de, and ea oriented as indicated, the maximum flow has value 22. We can see this is maximum by considering the cut (8, T), with 8 = {s, e} and T = {a, b, d, e, i, t} whose capacity is Csa + Csd + Cee + Cej = 8 + 7 + 2 + 5 = 22. a 13,10 b a (a) (b) s t b 3,3 c c 7. (a) Trees! A graph is a tree if and only if there is precisely one path between any two vertices. (See Proposition 12.1.2.) (b) Cycles! To see this, note that a Hamiltonian graph g has a cycle containing all the vertices of g, so there are at least two edge disjoint paths between any two vertices (either way round). If, in addition to the Hamiltonian cycle, there was another edge uv, then there would be three edge disjoint paths between u and v (edge uv and either way round the cycle). So the graphs with the given property are precisely those which are cycles....
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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