Discrete Mathematics with Graph Theory (3rd Edition) 388

Discrete Mathematics with Graph Theory (3rd Edition) 388 -...

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386 Solutions to Exercises Exercises 14.2 1. (a) i. [BB] One flow-augmenting chain is sbadt in which the slack is 1. ii. [BB] Here is a maximum flow, of value 7. We can see this is maximum by examining the cut S = {s,a,b}, T = {c,d,e,/,t}, of capacity Csc + Cad + Cbe = 3 + 3 + 1 = 7, the value of the flow. d f (b) i. One flow-augmenting chain is sadt, with slack 1. Another is sadcbt, also with slack 1. a 2,2 b ii. We show a maximum flow, of value 9. We can see that this is maximum by examining t s G-~~--~~~----~----~ the cut S = {s}, T = {a,b,c,d,e,t} which has capacity Csa + Cse + Csc = 4 + 3 + 2 = 9, the value of the flow. c 1,0 d t (c) i. One flow-augmenting chain is sagt, which has slack 1. Another is
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Unformatted text preview: sabce / gt, which has slack 1 as well. 11. We show a maximum flow, of value 8. We can see this is maximum by examining the cut s S = {s,a,b,c,e,/,g}, T = {d,t} which has capacity Ccd + Cfd + Cgt = 5 + 1 + 2, the value of the flow. (d) i. [BB] One flow-augmenting chain is saed/gt, which has slack 1. ii. [BB] We show a maximum flow, of value 20. We can see this is maximum by examining the cutS = {s,a,b,c}, T = {d,e,/,g,h,t}. This has capacity Cse+Csh +Ccd = 7+4+9 = 20, the value of the flow. a sCf-~---if-____ ""--"'--+--+--~t e 6,6 f 7,7 9 (e) 1. One flow-augmenting chain is sacghkt in which the slack is 4. ii. We show a maximum flow, of value 13. t...
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