## Abstract

We investigate the parameterized complexity of finding subgraphs with hereditary properties on graphs belonging to a hereditary graph class. Given a graph G, a non-trivial hereditary property Π and an integer parameter k, the general problem P(G, Π, k) asks whether there exists k vertices of G that induce a subgraph satisfying property Π. This problem, P(G, Π, k) has been proved to be NP -complete by Lewis and Yannakakis. The parameterized complexity of this problem is shown to be W[ 1 ] -complete by Khot and Raman, if Π includes all trivial graphs (graphs with no edges) but not all complete graphs and vice versa; and is fixed-parameter tractable, otherwise. As the problem is W[ 1 ] -complete on general graphs when Π includes all trivial graphs but not all complete graphs and vice versa, it is natural to further investigate the problem on restricted graph classes. Motivated by this line of research, we study the problem on graphs which also belong to a hereditary graph class and establish a framework which settles the parameterized complexity of the problem for various hereditary graph classes. In particular, we show that: P(G, Π, k) is solvable in polynomial time when the graph G is co-bipartite and Π is the property of being planar, bipartite or triangle-free (or vice-versa).P(G, Π, k) is fixed-parameter tractable when the graph G is planar, bipartite or triangle-free and Π is the property of being planar, bipartite or triangle-free, or graph G is co-bipartite and Π is the property of being co-bipartite.P(G, Π, k) is W[ 1 ] -complete when the graph G is C_{4} -free, K_{1, 4} -free or a unit disk graph and Π is the property of being either planar or bipartite.

Original language | English |
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Title of host publication | Fundamentals of Computation Theory - 23rd International Symposium, FCT 2021, Proceedings |

Editors | Evripidis Bampis, Aris Pagourtzis |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 217-229 |

Number of pages | 13 |

ISBN (Print) | 9783030865924 |

DOIs | |

State | Published - 1 Jan 2021 |

Event | 23rd International Symposium on Fundamentals of Computation Theory, FCT 2021 - Virtual, Online Duration: 12 Sep 2021 → 15 Sep 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 | 12867 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 23rd International Symposium on Fundamentals of Computation Theory, FCT 2021 |
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City | Virtual, Online |

Period | 12/09/21 → 15/09/21 |

## Keywords

- Fixed-parameter tractable
- Hereditary properties
- Ramsey’s theorem
- W-hardness

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)