ps3 - 6.889: Algorithms for Planar Graphs and Beyond...

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6.889: Algorithms for Planar Graphs and Beyond September 28, 2011 Problem Set 3 This problem set is due Wednesday, October 5 at noon. 1. Let P := { p 1 ,...,p k } be points in the plane and { Q 1 ,...,Q t } be a partition of P into t sets. Argue (informally) that there exist disjoint curves in the plane, C 1 ,...,C t , such that for i = 1 ,...,t , Q i C i . Deduce (informally) that there is a suitable function g , such that for every k , the planar grid graph G of size g ( k ) × g ( k ) has the following property: if u 1 ,...,u k V ( G ) are sufficiently far apart from each other and from the boundary (i.e. their pairwise distance and distance to the boundary is at least f ( k ) for a suitable function f ) and { Q 1 ,...,Q t } is a partition of { u 1 ,...,u k } , then there exist disjoint trees T 1 ,...,T t in G such that for i = 1 ,...,t , Q i T i . 2. For a vertex v in a graph G and a permutation π v of its neighbors, define the operation split ( v,π v ) as replacing v by a path P v of length degree( v ) and connecting each of
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