midtermsoln-1

midtermsoln-1 - CME 305: Discrete Mathematics and...

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Unformatted text preview: CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi (saberi@stanford.edu) Midterm 02/28/10 Problem 1. Show that a graph has a unique minimum spanning tree if, for every cut of the graph, the edge with the smallest cost across that cut is unique. Show that the converse is not true by giving a counterexample. Solution: Suppose MST is not unique, i.e., there exist T 1 and T 2 where both of them are MST and they are not identical. Suppose e 1 T 1 but e 1 / T 2 , if we remove e 1 from T 1 , then we will have two trees with vertex sets V 1 and V 2 . By problem 1 of HW #2, we know that e 1 is a minimum cost edge in the cut between V 1 and V 2 . Now consider T 2 , again problem 1 of HW #2, we know that T 2 contains an edge e 2 that is a minimum cost edge of the cut between V 1 and V 2 . However, since e 2 6 = e 1 we must have: c ( e 1 ) = c ( e 2 ) which is contradicting with the assumption that for every cut of the graph, the edge with the smallest cost across that cut is unique. Counter-example for the converse: suppose the graph is a tree with 3 nodes, where both edges have cost 1. Let v be the node with degree two. Clearly the only spanning tree is the graph itself, but the cut between...
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midtermsoln-1 - CME 305: Discrete Mathematics and...

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