Golovach, P.A. and Paulusma, D. and van Leeuwen, E.J. (2022) 'Induced disjoint paths in AT-free graphs.', *Journal of computer and system sciences.*, 124 . pp. 170-191.

## Abstract

Paths P1, . . . , Pk in a graph G = (V, E) are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (si, ti) contains k mutually induced paths Pi such that each Pi connects si and ti. This is a classical graph problem that is NP-complete even for k = 2. We study it for AT-free graphs. Unlike its subclasses of permutation graphs and cocomparability graphs, the class of AT-free graphs has no known characterisation by a geometric intersection model. However, by a new, structural analysis of the behaviour of Induced Disjoint Paths for AT-free graphs, we prove that it can be solved in polynomial time for AT-free graphs even when k is part of the input. This is in contrast to the situation for other well-known graph classes, such as planar graphs, claw-free graphs, or more recently, (theta,wheel)-free graphs (JCTB 2021), for which such a result only holds if k is fixed. As a consequence of our main result, the problem of deciding if a given ATfree graph contains a fixed graph H as an induced topological minor admits a polynomial-time algorithm. In addition, we show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard with parameter |VH|, even on a subclass of AT-free graphs, namely cobipartite graphs. We also show that the problems k-in-a-Path and k-in-a-Tree are polynomialtime solvable on AT-free graphs even if k is part of the input. These problems are to test if a graph has an induced path or induced tree, respectively, spanning k given vertices.

Item Type: | Article |
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Full text: | Publisher-imposed embargo until 29 October 2022. (AM) Accepted Manuscript Available under License - Creative Commons Attribution Non-commercial No Derivatives 4.0. File format - PDF (483Kb) |

Status: | Peer-reviewed |

Publisher Web site: | https://doi.org/10.1016/j.jcss.2021.10.003 |

Publisher statement: | © 2021 This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ |

Date accepted: | 18 October 2021 |

Date deposited: | 25 October 2021 |

Date of first online publication: | 29 October 2021 |

Date first made open access: | 29 October 2022 |

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