Finding Induced Paths of Given Parity in Claw-Free Graphs

Pim van 't Hof, Marcin Kamiński, Daniël Paulusma
2010 Algorithmica  
Citation for published item: rof vn 9tD F nd umi¡ nskiD wF nd ulusmD hniel @PHIPA 9pinding indued pths of given prity in lwEfree grphsF9D elgorithmiFD TP @IEPAF ppF SQUESTQF Further information on publisher's website: httpXGGdxFdoiForgGIHFIHHUGsHHRSQEHIHEWRUHES Publisher's copyright statement: The nal publication is available at Springer via http://dx.The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal
more » ... research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Abstract. The Parity Path problem is to decide if a given graph contains both an induced path of odd length and an induced path of even length between two specified vertices. In the related problems Odd Induced Path and Even Induced Path, the goal is to determine whether an induced path of odd, respectively even, length between two specified vertices exists. Although all three problems are NP-complete in general, we show that they can be solved in O(n 5 ) time for the class of claw-free graphs. Two vertices s and t form an even pair in G if every induced path from s to t in G has even length. Our results imply that the problem of deciding if two specified vertices of a claw-free graph form an even pair, as well as the problem of deciding if a given claw-free graph has an even pair, can be solved in O(n 5 ) time and O(n 7 ) time, respectively. We also show that we can decide in O(n 7 ) time whether a claw-free graph has an induced cycle of given parity through a specified vertex. Finally, we show that a shortest induced path of given parity between two specified vertices of a claw-free perfect graph can be found in O(n 7 ) time. An extended abstract of this paper has been
doi:10.1007/s00453-010-9470-5 fatcat:emwqsmiedbfrnc4qd35kei5xki