Discrete Mathematics with Graph Theory (3rd Edition) 320

Discrete Mathematics with Graph Theory (3rd Edition) 320 -...

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318 Chapter 11 Review 1. There are four odd vertices, labeled A, B, C, D in the figures to the right. In the unweighted case, at least two additional edges are required; in fact, two suffice, as shown. The table shows the calculations required for the weighted graph. It is necessary to duplicate four edges, as shown. Solutions to Review Exercises ABC CffP 3 3 1 Weighted Case Partition into pairs Sum of lengths of shortest paths {A,B},{C,D} {A, C}, {B, D} {A,D},{B,C} 3+4=7 4+3=7 4+1=5 2. This graph has exactly two vertices of odd degree, C and G. The shortest path from C to G is CADG, of length 8. (This follows from Exercise 27 of the Review Problems for Chapter 10; it also follows quickly by inspection.) Hence, we solve the problem by duplicating edges CA, AD and DG. 3. Let us label the bipartition sets {UI, U2, U3} and {v!, V2, V3, V4, V5, V6}. Our graph has six vertices of odd degree, namely, VI, •. . ,V6. The shortest path between any two of these will always be of length two, passing through one of
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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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