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ps9 - Introduction to Algorithms Massachusetts Institute of...

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Introduction to Algorithms December 6, 2002 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Shafi Goldwasser Handout 31 Problem Set 9 (Optional) This problem set is not due; it is optional. Reading: Chapter 34 Problem 9-1. Prove that the following problems are in : (a) there exists a simple path in between and , of length at least (b) it is possible to assign one of three “colors” to each vertex of such that no two neighboring vertices are assigned the same color Problem 9-2. A subgraph of a graph is a graph where ; i.e. it is a subset of the vertices together with all the edges of the original graph which are incident to these vertices. Consider the problem L ARGEST C OMMON S UBGRAPH : Given two graphs and and an inte- ger , determine whether there is a graph with edges which is a subgraph of both and . ( Hint: Reduce from C LIQUE .) Problem 9-3. A perfect matching in an undirected graph is a collection of edges such that each node has exactly one edge of incident to it. (In other words, the degree of each
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