ps1 - 6.889: Algorithms for Planar Graphs and Beyond...

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6.889: Algorithms for Planar Graphs and Beyond September 14, 2011 Problem Set 1 This problem set is due Thursday, September 22 at noon. 1. Sparsity Lemma: Prove that for a planar a embedded graph in which every face has size at least three, m 3 n - 6, where m is the number of edges and n is the number of vertices. 2. Minimum Spanning Tree: Give a linear-time algorithm for minimum-weight spanning tree in a connected planar graph. Refer to the textbook for a discussion of representing embedded graphs in computations and efficiently performing basic operations such as contractions and deletions. You are encouraged to use the following results, which you need not prove. Let G be a graph with edge-weights, and let v be a vertex. Let
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