Universal Slope Sets for 1-Bend Planar Drawings

We prove that every set of -1 slopes is 1-bend universal for the planar graphs with maximum vertex degree . This means that any planar graph with maximum degree 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 -1 slopes. Our result improves over previous literature in three ways: Firstly, it improves the known upper bound of 1) on the 1-bend planar slope number; secondly, the previously known algorithms compute 1-bend planar drawings by using sets of O() 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 which is worst-case optimal up to a factor of . Our proofs are constructive and give rise to a linear-time drawing algorithm.