Discrete Mathematics with Graph Theory (3rd Edition) 357

Discrete Mathematics with Graph Theory (3rd Edition) 357 -...

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Chapter 12 355 (b) No. The graph at the right has ~(n - l)(n - 2) + 1 edges but also a bridge. 14. (a) This is clear if the algorithm starts backtracking immediately after labeling k since edge ik was used to label k and immediately afterwards in the backtracking. Suppose the algorithm continues to label vertices after k (because there are unlabeled vertices adjacent to k at the time label k is used. The algorithm assigns k + 1 to a vertex adjacent to k, and continues to assign labels k + i, i > 0, until, at some point, after assigning k + s, s > 0, the algorithm backtracks through vertices all of whose adjacent vertices have already been labeled to a vertex with label t ::::; k. Let t be the maximal such integer. The result follows if we can show that t = k (since after the backtracking reaches k, it backtracks directly to i). But t = k follows by a straightforward induction argument. The result is clear if s = 1. Assume s > I, 1 ::::;
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Unformatted text preview: r < s and the result is true for all such r. The vertex with label k + s was labeled from a vertex k + r, 0 ::::; r < s. If r = 0, the backtracking goes k + s --+ k --+ i. If r > 0, the backtracking goes k + s --+ k + r --+ k (by the induction hypothesis) --+ l. In either case,ik is used twice. (b) Solution 1. Vertex k gets labeled from i. By part (a), edge ik is used twice by the algorithm, once when k is labeled and once on a backtracking. Solution 2. The result is clear for k = I, so assume k > 1 and the result is true for all i, 1 ::::; i < k. Now k received its label from i, with 1 ::::; i < k and the algorithm backtracks to i (by the induction hypothesis). Since ki must be the last edge on such a backtracking k also appears in this backtracking. 15. (a) [BB] Orient the depth first search spanning tree as in 12.6.4. (b) [BB] This follows from the definitions. Chapter 12 Review 1. There are 22 such subsequences in all. start...
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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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