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ps1sol - 6.889 Algorithms for Planar Graphs and Beyond...

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6.889: Algorithms for Planar Graphs and Beyond Problem Set 1 - Solutions 1. Solution: Since every face has size at least three, and each edge is in exactly two faces, 3 f 2 m (here, f is the number of faces). Substituting into Euler’s formula we get 2 = n - m + f n - m/ 3. Or, equivalently, m 3 n - 6. 2. Solution: The crucial observation is that in any planar graph with no self loops and no parallel edges, there always exists a node whose degree is at most 5. To see this, note that by the sparsity of planar graphs, v V degree( v ) = 2 m 6 n - 12. The algorithm is: 1: T ← ∅ 2: while G is not empty do 3: choose an arbitrary node v s.t. deg ( v ) 5 4: let e be the minimal weight edge incident to v 5: T T ∪ { e } 6: contract e , eliminating any parallel edges that occur by only keeping the one with minimal weight. 7: end while First, we may assume that G contains no self loops and parallel edges since we can detect and delete them in linear time (for parallel edges delete all but the lightest edge). The correctness
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