MIT6_042JS10_lec20_prob - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science March 19 Prof. Albert R. Meyer revised March 12, 2010, 1326 minutes In-Class Problems Week 7, Fri. Problem 1. Figures 1–4 show different pictures of planar graphs. 1 a b c d Figure 1 a b c d Figure 2 a b c d e Figure 3 a b c d e Figure 4 Creative Commons 2010, Prof. Albert R. Meyer .
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2 In-Class Problems Week 7, Fri. (a) For each picture, describe its discrete faces (simple cycles that define the region borders). (b) Which of the pictured graphs are isomorphic? Which pictures represent the same planar em- bedding ? – that is, they have the same discrete faces. (c) Describe a way to construct the embedding in Figure 4 according to the recursive Defini- tion 12.3.1 of planar embedding. For each application of a constructor rule, be sure to indicate the faces (cycles) to which the rule was applied and the cycles which result from the application. Problem 2. Prove the following assertions by structural induction on the definition of planar embedding. (a) In a planar embedding of a graph, each edge is traversed a total of two times by the faces of the embedding. (b)
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MIT6_042JS10_lec20_prob - Massachusetts Institute of...

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