## Abstract

Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects induced by elementary, local changes. A natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other?

We consider flip graphs on so-called α-orientations of a graph G, in which every vertex v has a specified outdegree α(v) and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two α-orientations of a planar graph G is at most 2 is NP-complete. This also holds in the special case of plane perfect matchings, where flips involve alternating cycles. We also consider the dual question of the flip distance between graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard, but if we only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Subtitle of host publication | 45th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2019; Catalonia; Spain; 19 June 2019 through 21 June 2019 |

Editors | Ignasi Sau, Dimitrios M. Thilikos |

Place of Publication | Switzerland |

Publisher | Springer-Verlag London Ltd. |

Pages | 120-134 |

Number of pages | 15 |

Volume | 11789 |

ISBN (Electronic) | 9783030307868 |

ISBN (Print) | 9783030307851 |

DOIs | |

Publication status | Published - 12 Sep 2019 |

Externally published | Yes |

Event | 45th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2019 - Catalonia, Spain Duration: 19 Jun 2019 → 21 Jun 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Publisher | Springer Nature Switzerland |

Volume | 11789 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 45th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2019 |
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Country | Spain |

City | Catalonia |

Period | 19/06/19 → 21/06/19 |

## Keywords

- Flip distance
- Graph orientation
- α-orientation