Discrete Mathematics with Graph Theory (3rd Edition) 337

Discrete Mathematics with Graph Theory (3rd Edition) 337 -...

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Section 12.3 335 connected to A via two paths of edges and, as we have noted, Dijkstra's algorithm provides just one path. Also note that if a vertex X is encountered on the shortest path from A to B, as given by the algorithm, then the shortest path from A to X given by the algorithm agrees with the first part of the shortest path from A to B. Exercises 12.3 1. (a) [BB] We want five edges (since there are six vertices). Choose BC, then AD, FE, and DE. We would like next to choose AE, but this would complete a circuit with AD and DE, so we choose AC and obtain the spanning tree shown, of weight 13. (c) (a) Bl0[)A: (b) ~3~2A 35 2~ ~ A 2 ~3 S BET C E C E 1 2 2 6 3 C F D D (b) As in part (a), we need five edges. Choose AD, then BC, then EF, then AE. Now DE is no good, so AC will finish the job. The weight of a minimum spanning tree is 122 as shown. (c) We want seven edges (since there are eight vertices). First choose SC, then CE, then AD, then BC and DT. Now BE cannot be chosen since it would complete a circuit with previously chosen
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