Abstract
We consider the problem of orienting the edges of a planar graph in such a way that the out-degree of each vertex is minimized. If, for each vertex v, the out-degree is at most d, then we say that such an orientation is d-bounded. We prove the following results: • Each planar graph has a 5-bounded acyclic orientation, which can be constructed in linear time. • Each planar graph has a 3-bounded orientation, which can be constructed in linear time. • A 6-bounded acyclic orientation, and a 3-bounded orientation, of each planar graph can each be constructed in parallel time O(log n log ∗ n) on an EREW PRAM, using O( n/log n log ∗ n) processors. As an application of these results, we present a data structure such that each entry in the adjacency matrix of a planar graph can be looked up in constant time. The data structure uses linear storage, and can be constructed in linear time.
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