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COT_6410_Homework_2 - COT 6410 Homework#2 Due next Tuesday...

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COT 6410 Homework #2. Due next Tuesday in class (June 5, 2007) Consider the following decision problem: Disjoint Clique Cover (DCC) Given: a graph G = (V, E) and an integer K. Question: Can V be partitioned into k ≤ K sets, V 1 , V 2 , …, V k where each set forms a complete subgraph in G? Find (and demonstrate the validity of) a decision problem π for which there is a polynomial transformation to, and from, DCC. {Hint: we have seen this problem.} Solution: Use the Graph k-Colorability Problem. Graph K-Color Problem (COL) Given: a graph G and in integer k. Question: Can G be properly colored using at most k colors? "properly" means the end points of each edge in G must have a different color. COL DCC 1) Construct the instance of DCC from an arbitrary instance of COL. Let (G, k) be an arbitrary instance I COL of COL. Construct the complement graph G' of G and let the k of DCC be the k of COL. Thus, the constructed instance f(I COL ) = I DCC of DCC is (G', k). Since this takes order the number of edges of G time, and the number of edges is O(n 2
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