## Abstract

Classical vertex subset problems demanding connectivity are of the following form: given an input graph G on n vertices and an integer k, find a set S of at most k vertices that satisfies a property and G[S] is connected. In this paper, we initiate a systematic study of such problems under a specific connectivity constraint, from the viewpoint of Kernelization (Parameterized) Complexity. The specific form that we study does not demand that G[S] is connected but rather G[S] has a closed walk containing all the vertices in S. In particular, we study Closed Walk-Subgraph Vertex Cover (CW-SVC, in short), where given a graph G, a set X⊆ E(G), and an integer k; the goal is to find a set of vertices S that hits all the edges in X and can be traversed by a closed walk of length at most k in G. When X is E(G), this corresponds to Closed Walk-Vertex Cover (CW-VC, in short). One can similarly define these variants for Feedback Vertex Set, namely Closed Walk-Subgraph Feedback Vertex Set (CW-SFVS, in short) and Closed Walk-Feedback Vertex Set (CW-FVS, in short). Our results are as follows: CW-VC and CW-SVC both admit a polynomial kernel, in contrast to Connected Vertex Cover that does not admit a polynomial kernel unless NP⊆ coNP/ poly.CW-FVS admits a polynomial kernel. On the other hand CW-SFVS does not admit even a polynomial Turing kernel unless the polynomial-time hierarchy collapses. We complement our kernelization algorithms by designing single-exponential FPT algorithms – 2 ^{O} ^{(} ^{k} ^{)}n^{O} ^{(} ^{1} ^{)} – for all the problems mentioned above.

Original language | English |
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Title of host publication | Algorithms and Complexity - 12th International Conference, CIAC 2021, Proceedings |

Editors | Tiziana Calamoneri, Federico Corò |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 300-313 |

Number of pages | 14 |

ISBN (Print) | 9783030752415 |

DOIs | |

State | Published - 1 Jan 2021 |

Externally published | Yes |

Event | 12th International Conference on Algorithms and Complexity, CIAC 2021 - Virtual, Online Duration: 10 May 2021 → 12 May 2021 |

### 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 | 12701 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 12th International Conference on Algorithms and Complexity, CIAC 2021 |
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City | Virtual, Online |

Period | 10/05/21 → 12/05/21 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science