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Induced disjoint paths in claw-free graphs.

Golovach, P.A. and Paulusma, D. and van Leeuwen, E.J. (2015) 'Induced disjoint paths in claw-free graphs.', SIAM journal on discrete mathematics., 29 (1). pp. 348-375.


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 specied vertices (si; ti) contains k mutually induced paths Pi such that Pi connects si and ti for i = 1; : : : ; k. We show that this problem is xed-parameter tractable for claw-free graphs when parameterized by k. Several related problems, such as the k-in-a-Path problem, are proven to be xed-parameter tractable for claw-free graphs as well. We show 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 clawfree 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:Article
Keywords:Induced disjoint paths, Claw-free graphs, Parameterized complexity
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Publisher statement:© 2015 Society for Industrial and Applied Mathematics
Date accepted:No date available
Date deposited:13 February 2015
Date of first online publication:2015
Date first made open access:No date available

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