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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 UI, U2, U3.
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