### Abstract

A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (reconfiguration) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. This gives a linear-time algorithm for finding an actual shortest reconfiguration sequence for interval graphs. Since interval graphs are exactly the graphs that are simultaneously chordal and co-comparability, we then complement the positive result by showing that even deciding reachability is PSPACE-complete for chordal graphs and for co-comparability graphs. The hardness for chordal graphs holds even for split graphs. We also consider the case where c is a fixed constant and show that in such a case the reachability problem is polynomial-time solvable for split graphs but still PSPACE-complete for co-comparability graphs. The complexity of this case for chordal graphs remains unsettled. As by-products, our positive results give the first polynomial-time solvable cases (split graphs and interval graphs) for Feedback Vertex Set Reconfiguration.

Original language | English |
---|---|

Title of host publication | 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018 |

Editors | David Eppstein |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 271-2713 |

Number of pages | 2443 |

ISBN (Electronic) | 9783959770682 |

DOIs | |

Publication status | Published - 2018 Jun 1 |

Event | 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018 - Malmo, Sweden Duration: 2018 Jun 18 → 2018 Jun 20 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 101 |

ISSN (Print) | 1868-8969 |

### Other

Other | 16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018 |
---|---|

Country | Sweden |

City | Malmo |

Period | 18/6/18 → 18/6/20 |

### Keywords

- Colorable set
- Perfect graph
- Phrases reconfiguration

### ASJC Scopus subject areas

- Software

## Fingerprint Dive into the research topics of 'Reconfiguration of colorable sets in classes of perfect graphs'. Together they form a unique fingerprint.

## Cite this

*16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018*(pp. 271-2713). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 101). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SWAT.2018.27