# Level-Planar Drawings with Few Slopes

## TL;DR

This paper studies level-planar straight-line graph drawings with a limited number of slopes, providing efficient algorithms for proper graphs and complexity results for non-proper graphs.

## Contribution

It introduces algorithms for constructing level-planar drawings with few slopes and analyzes the computational complexity of related extension and simultaneous drawing problems.

## Key findings

- Efficient $O(n ext{ log}^2 n / ext{log} ext{log} n)$-time algorithm for proper level graphs.
- Algorithms for extension and simultaneous drawing problems with $O(n^{4/3} ext{ log} n)$ and $O( ext{lambda} n^{10/3} ext{ log} n)$ complexity.
- NP-hardness results for non-proper level graphs with few slopes.

## Abstract

We introduce and study level-planar straight-line drawings with a fixed number $\lambda$ of slopes. For proper level graphs, 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 $O({\lambda} n^{10/3} \log n)$-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 $\lambda$ slopes is $\textsf{NP}$-hard even in restricted cases.