Golovach, P.A. and Paulusma, Daniel and van Leeuwen, E.J. (2012) 'Induced disjoint paths in claw-free graphs.', in *Algorithms, 20th Annual European Symposium, ESA 2012, Ljubljana, Slovenia, 10-12 September 2012 ; proceedings.* Berlin ; Heidelberg: Springer, pp. 515-526. Lecture notes in computer science. (7501).

## Abstract

Paths P1,…,Pk in a graph G = (V,E) are said to be mutually induced if for any 1 ≤ i < j ≤ k, Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to test whether a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that Pi connects si and ti for i = 1,…,k. This problem is known to be NP-complete already for k = 2, but for n-vertex claw-free graphs, Fiala et al.gave an nO(k)-time algorithm. We improve the latter result by showing that the problem is fixed-parameter tractable for claw-free graphs when parameterized by k. Several related problems, such as the k-in-a-Path problem, are shown to be fixed-parameter tractable for claw-free graphs as well. We prove that an improvement of these results in certain directions is unlikely, for example by noting that the Induced Disjoint Paths problem cannot have a polynomial kernel for line graphs (a type of claw-free graphs), unless NP ⊆ coNP/poly. Moreover, the problem becomes NP-complete, even when k = 2, for the more general class of K1,4-free graphs. Finally, we show that the nO(k)-time algorithm of Fiala et al.for testing whether a claw-free graph contains some k-vertex graph H as a topological induced minor is essentially optimal by proving that this problem is W[1]-hard even if G and H are line graphs.

Item Type: | Book chapter |
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Full text: | Full text not available from this repository. |

Publisher Web site: | http://dx.doi.org/10.1007/978-3-642-33090-2_45 |

Date accepted: | No date available |

Date deposited: | No date available |

Date of first online publication: | 2012 |

Date first made open access: | No date available |

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