ps5 - 6.889: Algorithms for Planar Graphs and Beyond...

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6.889: Algorithms for Planar Graphs and Beyond October 12, 2011 Problem Set 5 This problem set is due Wednesday, October 19 at noon. 1. For m > 2 k , let G be an m × m grid, and G 0 be the central ( m - 2 k ) × ( m - 2 k ) subgrid of G . Let N be a set of at least k 4 vertices in G 0 . (a) Show that one can designate a side of G as the x -axis and the other one as the y -axis in such a way that at least k 2 vertices of N have different y -coordinates. (b) Let N 0 = { v 1 ,...,v k 2 } ⊆ N be a set of exactly k 2 vertices with different y - coordinates and assume that the v j are sorted by increasing y -coordinate. For 0 i < k , let N i = { v j : ki j < k ( i + 1) } . Show (essentially by picture) that there exist k disjoint paths in G such that each path contains exactly one vertex out of each of N 0 ,...,N k - 1 . (c) Show that G contains a model of a k × k -grid K in which the image of each vertex of K in G contains exactly one vertex of N 0 . 2. Let
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