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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 lineartime algorithm for minimumweight 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 edgeweights, and let
v
be a vertex. Let
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 Fall '11
 ErikDemaine
 Algorithms

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