Induced disjoint paths in claw-free graphs

Golovach, P.A.; Paulusma, D.; van Leeuwen, E.J.

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 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.

Citation

Golovach, P., Paulusma, D., & van Leeuwen, E. (2015). Induced disjoint paths in claw-free graphs. SIAM Journal on Discrete Mathematics, 29(1), 348-375. https://doi.org/10.1137/140963200

• Journal Article Type: Article
• Publication Date: Jan 1, 2015
• Deposit Date: Dec 20, 2014
• Publicly Available Date: Feb 13, 2015
• Journal: SIAM Journal on Discrete Mathematics
• Print ISSN: 0895-4801
• Electronic ISSN: 1095-7146
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 29
• Issue: 1
• Pages: 348-375
• DOI: https://doi.org/10.1137/140963200
• Keywords: Induced disjoint paths, Claw-free graphs, Parameterized complexity
• Public URL: https://durham-repository.worktribe.com/output/1447883