Universal Slope Sets for 1-Bend Planar Drawings

Abstract

We prove that every set of $$\varDelta -1$$ slopes is 1-bend universal for the planar graphs with maximum vertex degree $$\varDelta $$ . This means that any planar graph with maximum degree $$\varDelta $$ admits a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges can be chosen in any given set of $$\varDelta -1$$ slopes. Our result improves over previous literature in three ways: Firstly, it improves the known upper bound of $$\frac{3}{2} (\varDelta -1)$$ on the 1-bend planar slope number; secondly, the previously known algorithms compute 1-bend planar drawings by using sets of $$O(\varDelta )$$ slopes that may vary depending on the input graph; thirdly, while these algorithms typically minimize the slopes at the expenses of constructing drawings with poor angular resolution, we can compute drawings whose angular resolution is at least $$\frac{\pi }{\varDelta -1}$$ , which is worst-case optimal up to a factor of $$\frac{3}{4}$$ . Our proofs are constructive and give rise to a linear-time drawing algorithm.