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MIT6_042JS10_lec20_sol - 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 22, 2010, 587 minutes Solutions to 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 Solutions to In-Class Problems Week 7, Fri. (a) For each picture, describe its discrete faces (simple cycles that define the region borders). Solution. Figs 1 & 2: abda, bcdb, abcda. Fig 3: abcdea, adea,abda,bcdb. Fig 4: abcda, abdea, bdcb, adea. (b) Which of the pictured graphs are isomorphic? Which pictures represent the same planar em- bedding ? – that is, they have the same discrete faces. Solution. Figs 1 & 2 have the same faces, so are different pictures of the same planar drawing. Figs 3 & 4 both have four faces, but they are different, for example, Fig 3 has a face with 5 edges, but the longest face in Fig 4 has 4 edges. (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. Solution. Here’s one way. (By Lemma 12.7.1 , the constructor steps could be done in any order.) recursive step faces vertex a (base case) a vertex b (base) b a b (bridge) aba vertex c (base) c b c (bridge) abcba vertex d (base) d c d (bridge) abcdcba a d (split) dabcd, dabcd b d (split) dabd, dbcd, abcda vertex e (base) e d e (bridge) dedabd, dbcd, abcda a e (split) abdea, adea, dbcd, abcda 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. Solution. Proof. The induction hypothesis is that if E is a planar embedding of a graph, then each edge is traversed exactly twice by the faces of E . Base case: There is one vertex and no edges, so this case holds vacuously.
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