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ps4sol - 6.889 Algorithms for Planar Graphs and Beyond...

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6.889: Algorithms for Planar Graphs and Beyond Problem Set 4 - Solutions 1. Solution: (a) Let ( T, ( B t ) t V ( T ) ) be a tree decomposition of G . We show that there exists a bag B z such that C B z . For any edge uv of T , define T u uv as the connected component of T - uv that contains u . Let W u be the union of all bags in T u uv and define W v analogously. Now notice that at least one of W u and W v must contain all the vertices of C : for otherwise, there were vertices x, y C such that x appears only in W u and y appears only in W v , and thus there would be no bag that contains both of x and y – a contradiction. If C W u , direct the edge uv towards u ; otherwise direct it towards v . Since T is a tree, there is a vertex z V ( T ) such that all the edges incident to z are directed towards z . We claim that, C B z . Suppose not; let x C be a vertex not in B z . Consider a neighbor u of z such that x appears in the bags of T u zu . But since the edge zu is directed towards z , there must also be a bag in T z zu
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