Abstract
We study level planarity testing of graphs with a fixed combinatorial embedding for three different notions of combinatorial embeddings, namely the level embedding, the upward embedding and the planar embedding. These notions allow for increasing degrees of freedom in their corresponding drawings. For the fixed level embedding there are known and easy to test level planarity criteria. We use these criteria to prove an "untangling" lemma that plays a key role in a simple level planarity test for the case where only the upward embedding is fixed. This test is then adapted to the case where only the planar embedding is fixed. Further, we characterize radial upward planar embeddings, which lets us extend our results to radial level planarity. No algorithms were previously known for these problems.
Highlights
Level planarity and upward planarity are two natural planarity notions for directed graphs that enrich the notion of planarity of ordinary graphs by imposing additional requirements based on the directions of the edges
We have studied the upward and level planarity testing problems with fixed embedding for three notions of fixed embeddings, namely embeddings, upward embeddings and level embeddings
We provided a new combinatorial characterization of radial upward planarity with fixed embedding, which can be tested in O(n log3 n) time
Summary
Level planarity and upward planarity are two natural planarity notions for directed graphs that enrich the notion of planarity of ordinary graphs by imposing additional requirements based on the directions of the edges. Equivalence classes of such drawings can be combinatorially described by so-called embeddings,which give the circular order of edges around each vertex, upward embeddings, which give the linear orders of incoming and outgoing edges around each vertex, and level embeddings, which speficy for each level the order of its vertices and the edges that cross it These notions make sense in the radial setting, except that there level embeddings specify circular orderings rather than linear ones. We consider the level planar setting, finding that in this setting we can “untangle” drawings to insert additional edges This lets us devise linear-time algorithms for (radial) level planarity testing with fixed embedding.
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