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Independent feedback vertex set for P5-free graphs.

Bonamy, Marthe and Dabrowski, Konrad K. and Feghali, Carl and Johnson, Matthew and Paulusma, Daniel (2018) 'Independent feedback vertex set for P5-free graphs.', Algorithmica., 81 (4). pp. 1416-1449.


The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer k≥0 , to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding whether or not a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for P4 -free graphs. We show that it remains polynomial-time solvable for P5 -free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks whether or not a graph has an independent odd cycle transversal of size at most k for a given integer k≥0 . Finally, in line with our underlying research aim, we compare the complexity of Independent Feedback Vertex Set for H-free graphs with the complexity of 3-Colouring, Independent Odd Cycle Transversal and other related problems.

Item Type:Article
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Publisher statement:© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Date accepted:16 June 2018
Date deposited:18 June 2018
Date of first online publication:26 June 2018
Date first made open access:29 June 2018

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