## Abstract

We consider the problem of untangling a given (non-planar) straight-line circular drawing δ_{G} of an outerplanar graph G=(V,E) into a planar straight-line circular drawing of G by shifting a minimum number of vertices to a new position on the circle. For an outerplanar graph G, it is obvious that such a crossing-free circular drawing always exists and we define the circular shifting number shift^{∘}(δ_{G}) as the minimum number of vertices that are required to be shifted in order to resolve all crossings of δ_{G}. We show that the problem CIRCULAR UNTANGLING, asking whether shift^{∘}(δ_{G})≤K for a given integer K, is NP-complete. For n-vertex outerplanar graphs, we obtain a tight upper bound of shift^{∘}(δ_{G})≤n−⌊n−2⌋−2. Moreover, we study the CIRCULAR UNTANGLING for almost-planar circular drawings, in which a single edge is involved in all of the crossings. For this problem, we provide a tight upper bound [Formula presented] and present an O(n^{2})-time algorithm to compute the circular shifting number of almost-planar drawings.

Original language | English |
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Article number | 101975 |

Journal | Computational Geometry: Theory and Applications |

Volume | 111 |

DOIs | |

State | Published - 1 Apr 2023 |

Externally published | Yes |

## Keywords

- NP-hardness
- Outerplanarity
- Permutations and combinations
- Straight-line Graph drawing
- Untangling

## ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics