Level-Planar Drawings with Few Slopes

We introduce and study level-planar straight-line drawings with a fixed number 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)-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)-time and 𝑂(𝜆𝑛$^{10/3}$ 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 𝜆 slopes is NP-hard even in restricted cases.