Discrete Mathematics with Graph Theory (3rd Edition) 386

Discrete Mathematics with Graph Theory (3rd Edition) 386 -...

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384 Solutions to Exercises 91 " {a, c} as a plane graph then with straight edges and a floor plan E E F ~ m CDF B H A G F A G Exercises 14.1 1. (a) [BB] At a, I: v fva = fsa = 2 and I: v fav = fac + fae = 2 + 0 = 2. At e, I:v fve = fae + fbe = o + 1 = 1 and I:v f ev = f ec + fed = 0 + 1 = 1. At d, I:v f vd = fbd + fed = 3 + 1 = 4 and I:v fdv = fdt = 4. (b) [BB] The value ofthe flow is 6. (c) [BB] The capacity of the cut is Cac + Cae + Cbe + Cbd = 3 + 1 + 4 + 3 = 11. (d) [BB] No. Arc dt is saturated. (e) [BB] The flow is not maximum. For instance, it can be increased by adding 1 to the flow in the arcs along sact. (f) [BB] (fac-fca)+(fae-fea)+(he-feb)+(fbd-fdb) = (2-0)+(0-0)+(1-0)+(3-0) = 6. 2. (a) At a, I:v f va = f sa = 6 and I:v f av = f ac + f ae = 3 + 3 = 6. At e, I: v f ve = f ae + f se + fbe = 3+3+2 = 8 and I:v fev = fec+fet+fed = 3+2+3 = 8. Atd, I:v fVd = fbd+fed = 3+3 = 6 and I: v fdv = fdt = 6. (b) [BB] The value of the flow is 14.
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Unformatted text preview: (c) The capacity of the cut is Cct + Cet + Cdt = 6 + 2 + 6 = 14. (d) The saturated arcs are se, sb, ae, et, ed, ct and dt. (e) The flow is maximum, by Corollary 14.1.6. (f) (fct -ftc) + (fet -f te ) + (fdt -ftd) = (6 - 0) + (2 - 0) + (6 - 0) = 14. 3. (a) At a, I:v fva = fsa = 11 and I:v fav = fac + fad = 3 + 8 = 11. At c, I: v fvc = fac = 3 and I:v f cv = f cd + f cf = 0 + 3 = 3. At d, I: v f vd = fad + fbd + f cd = 8 + 1 + 0 = 9 and I:v fdv = fdt = 9. (b) The value of the flow is 21. (c) [BB] Cac + Cbe + Cdt = 3 + 9 + 9 = 21. (d) The flow is maximum, by Corollary 14.1.6. (e) (fac-fca)+(fbe-feb)+(fdt-ftd)+(fdc-fcd) = (3-0)+(9-0)+(9-0)+(0-0)+(3-0) = 21. 4. (a) At a, I: v f va = f sa = 3 and I: v f av = f ab + f ac + fad = 1 + 1 + 1 = 3. At b, I: v f vb = f ab = 1 and I:v fbv = fbc + fbe + fbt = 1 + 0 + 0 = 1. At e, I:v f ve = fbe + f de = 0 + 3 = 3 and I:v fev = f et = 3....
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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.

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