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ps2sol - 6.889: Algorithms for Planar Graphs and Beyond...

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6.889: Algorithms for Planar Graphs and Beyond September 21, 2011 Problem Set 2 This problem set is due Wednesday, September 28 at noon. 1. Prove that any undirected planar graph G with non-negative edge weights can be transformed into an undirected planar graph G 0 with maximum degree 3 such that, for any u,v V ( G ), d G ( u,v ) = d G 0 ( f ( u ) ,f ( v )), where f : V ( G ) V ( G 0 ) maps vertices between G and G 0 ; and • | V ( G 0 ) | = O ( | V ( G ) | ). Solution: We replace each node u of degree d by a cycle C u on d nodes. Each edge uv is represented by (1) one node c uv on the cycle C u , (2) one node c vu on the cycle C v , and (3) an edge c uv c vu . The edge c uv c vu is assigned the weight of uv . The edges on the cycle have zero weight. f maps any node u to an arbitrary node of C u . Shortest-path distances are maintained. For each edge we introduce two nodes. The number of nodes in G 0 is thus O ( | E ( G ) | ) = O ( | V ( G ) | ).
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This note was uploaded on 01/20/2012 for the course CS 6.889 taught by Professor Erikdemaine during the Fall '11 term at MIT.

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