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
This is the end of the preview. Sign up
access the rest of the document.