Level-Planar Drawings with Few Slopes

Abstract

We introduce and study level-planar straight-line drawings with a fixed number $$\lambda $$ λ of slopes. For proper level graphs (all edges connect vertices of adjacent levels), we give an $$O(n \log ^2 n / \log \log n)$$ O ( n log 2 n / log log n ) -time algorithm that either finds such a drawing or determines that no such drawing exists. Moreover, we consider the partial drawing extension problem, where we seek to extend an immutable drawing of a subgraph to a drawing of the whole graph, and the simultaneous drawing problem, which asks about the existence of drawings of two graphs whose restrictions to their shared subgraph coincide. We present $$O(n^{4/3} \log n)$$ O ( n 4 / 3 log n ) -time and $$O(\lambda n^{10/3} \log n)$$ O ( λ n 10 / 3 log n ) -time algorithms for these respective problems on proper level-planar graphs. We complement these positive results by showing that testing whether non-proper level graphs admit level-planar drawings with $$\lambda $$ λ slopes is -hard even in restricted cases.