We prove that every set of Δ- 1 slopes is 1-bend universal for the planar graphs with maximum vertex degree Δ. This means that any planar graph with maximum degree Δ 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 Δ- 1 slopes. Our result improves over previous literature in three ways: Firstly, it improves the known upper bound of 32(Δ-1) on the 1-bend planar slope number; secondly, the previously known algorithms compute 1-bend planar drawings by using sets of O(Δ) 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 πΔ-1, which is worst-case optimal up to a factor of 34. Our proofs are constructive and give rise to a linear-time drawing algorithm.

### Universal Slope Sets for 1-Bend Planar Drawings

#### Abstract

We prove that every set of Δ- 1 slopes is 1-bend universal for the planar graphs with maximum vertex degree Δ. This means that any planar graph with maximum degree Δ 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 Δ- 1 slopes. Our result improves over previous literature in three ways: Firstly, it improves the known upper bound of 32(Δ-1) on the 1-bend planar slope number; secondly, the previously known algorithms compute 1-bend planar drawings by using sets of O(Δ) 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 πΔ-1, which is worst-case optimal up to a factor of 34. Our proofs are constructive and give rise to a linear-time drawing algorithm.
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2019
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11391/1448705`
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