6.889: Algorithms for Planar Graphs and BeyondProblem Set 4 - Solutions1.Solution:(a) Let (T,(Bt)t∈V(T)) be a tree decomposition ofG. We show that there exists a bagBzsuch thatC⊆Bz. For any edgeuvofT, defineTuuvas the connected component ofT-uvthat containsu. LetWube the union of all bags inTuuvand defineWvanalogously. Now notice that at leastone ofWuandWvmust contain all the vertices ofC: for otherwise, there were verticesx, y∈Csuch thatxappears only inWuandyappears only inWv, and thus there would be no bag thatcontains both ofxandy– a contradiction.IfC⊆Wu, direct the edgeuvtowardsu; otherwise direct it towardsv. SinceTis a tree, thereis a vertexz∈V(T) such that all the edges incident tozare directed towardsz. We claim that,C⊆Bz. Suppose not; letx∈Cbe a vertex not inBz. Consider a neighboruofzsuch thatxappears in the bags ofTuzu. But since the edgezuis directed towardsz, there must also be a baginTzzu
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