## Abstract

A vertex subset I of a graph G is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from I. The k -Path Vertex Cover Reconfiguration (k -PVCR) problem asks if one can transform one k-path vertex cover into another via a sequence of k-path vertex covers where each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. We investigate the computational complexity of k -PVCR from the viewpoint of graph classes under the well-known reconfiguration rules: TS, TJ, and TAR. The problem for k = 2, known as the Vertex Cover Reconfiguration (VCR) problem, has been well-studied in the literature. We show that certain known hardness results for VCR on different graph classes including planar graphs, bounded bandwidth graphs, chordal graphs, and bipartite graphs, can be extended for k -PVCR. In particular, we prove a complexity dichotomy for k -PVCR on general graphs: on those whose maximum degree is 3 (and even planar), the problem is PSPACE-complete, while on those whose maximum degree is 2 (i.e., paths and cycles), the problem can be solved in polynomial time. Additionally, we also design polynomial-time algorithms for k -PVCR on trees under each of TJ and TAR. Moreover, on paths, cycles, and trees, we describe how one can construct a reconfiguration sequence between two given k-path vertex covers in a yes-instance. In particular, on paths, our constructed reconfiguration sequence is shortest.

Original language | English |
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Title of host publication | WALCOM |

Subtitle of host publication | Algorithms and Computation - 14th International Conference, WALCOM 2020, Proceedings |

Editors | M. Sohel Rahman, Kunihiko Sadakane, Wing-Kin Sung |

Publisher | Springer |

Pages | 133-145 |

Number of pages | 13 |

ISBN (Print) | 9783030398804 |

DOIs | |

Publication status | Published - 2020 |

Event | 14th International Conference and Workshops on Algorithms and Computation, WALCOM 2020 - Singapore, Singapore Duration: 2020 Mar 31 → 2020 Apr 2 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12049 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 14th International Conference and Workshops on Algorithms and Computation, WALCOM 2020 |
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Country | Singapore |

City | Singapore |

Period | 20/3/31 → 20/4/2 |

## Keywords

- Combinatorial Reconfiguration
- Computational complexity
- PSPACE-completeness
- Polynomial-time algorithms
- k-path vertex cover

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)