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Discrete Mathematics with Graph Theory (3rd Edition) 351

# Discrete Mathematics with Graph Theory (3rd Edition) 351 -...

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Section 12.5 (c) The final backtracking is 17, 13, 1. (d) The final backtracking is 15, 14, 12, 11, 10, 3, 2, 1. 14 349 15 16 15 1 7 12 13 3 9 8 3. We use the strong form of mathematical induction on k, a vertex label, the result being clearly true for k = 1. Suppose k > 1 and the result is true for all f in the interval 1 ~ f < k; thus, there is a path from 1 to f which uses only edges required by the algorithm. We must prove the result for k: We must prove that there is a path from 1 to k which uses only edges required by the algorithm. Vertex k acquires its label because the algorithm finds this vertex unlabeled but adjacent to a previously labeled vertex f. Since at the time a vertex is labeled, the algorithm always picks the smallest unused label, we must have f < k. So, by the induction hypothesis, there is a path from 1 to f using edges required by the algorithm; also, the algorithm uses the edge fk to label k. Thus, there is a path from 1 to k along edges which the algorithm requires.
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