Discrete Mathematics with Graph Theory (3rd Edition) 343

Discrete Mathematics with Graph Theory (3rd Edition) 343 -...

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Section 12.3 341 ® A B~5 F 1 2 C E 6 3 C D D (c) [BB] As shown on the right, the minimum weight of a spanning tree after B is removed is 12. The two least edges at B have weights 1 and 5, so we obtain 12 + 1 + 5 = 18 as a lower bound. (d) It is not possible to improve the lower bound in (c), as we can see from the cycle found in (a). 16. (a) The graph is complete, so an obvious approach is to try to choose the lowest weight available edge at each vertex. Such a cycle is ADEF BGA, which has weight 2 + 36 + 21 + 55 + 13 + 51 = 178. (b) As shown on the right, the minimum weight of a spanning tree after D is removed is 120. The two least edges at D have weights 2 and 36, so we obtain 120 + 2 + 36 = 158 as a lower bound. A A BV035 Bq, 5A5 t;¥F 13 13~ ~21 C 2 E C E D (c) As shown on the left, the minimum weight of a spanning tree after F is removed is 101. The two edges of least weight at F have weights 21 and 52, so we obtain an estimate of 101 + 21 +52 = 174 as a lower bound for the weight of any Hamiltonian cycle. (d) As shown at the right, if vertex
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