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Discrete Mathematics with Graph Theory (3rd Edition) 397

# Discrete Mathematics with Graph Theory (3rd Edition) 397 -...

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Chapter 14 There is still another flow-augmenting chain, namely sbdait, of slack 3. The flow increases to that shown on the right. The flow has value 25. We claim that this is maximal. Constructing an (s, t)-cut using the method of Theorem 14.2.2 gives 8 = {s, b, d, a, e}, T = {e,i,t}. Now cap(8, T) = Cae + Caj + Cbe + Cej + Cdt = 4 + 3 + 2 + 5 + 11 = 25, agreeing with our value for the flow. 395 c f 4. We are given that the value of the flow is cap(8, T) = LUES,VET C UV = LVET C SV = LVEV c s ". However, val ( F) = LVEV f SV. Since f sv ~ C SV always, those can only be equal if f sv = C SV for all v, that is, each arc leaving s is saturated. 5. Add vertices s and t as shown. A maximal flow in this network is given. Note that all requirements have not been met. 8 E t H 6. (a) With arcs ba and be oriented as indicated, the maximum flow has value 9. We can see this is maximum by considering the cut (8, T), with 8 = {s,
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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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