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MIT6_042JS10_lec11_prob

MIT6_042JS10_lec11_prob - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science March 1 Prof. Albert R. Meyer revised February 27, 2010, 1329 minutes In-Class Problems Week 5, Mon. Problem 1. If a and b are distinct nodes of a digraph, then a is said to cover b if there is an edge from a to b and every path from a to b traverses this edge. If a covers b , the edge from a to b is called a covering edge . (a) What are the covering edges in the following DAG? 12 6 1 8 2 4 10 5 7 11 9 3 (b) Let covering ( D ) be the subgraph of D consisting of only the covering edges. Suppose D is a finite DAG. Explain why covering ( D ) has the same positive path relation as D . Hint: Consider longest paths between a pair of vertices. (c) Show that if two DAG ’s have the same positive path relation, then they have the same set of covering edges. (d) Conclude that covering ( D ) is the unique DAG with the smallest number of edges among all digraphs with the same positive path relation as D .
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