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Proper Interval Vertex Deletion

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Abstract

The NP-complete problem Proper Interval Vertex Deletion is to decide whether an input graph on n vertices and m edges can be turned into a proper interval graph by deleting at most k vertices. Van Bevern et al. (In: Proceedings WG 2010. Lecture notes in computer science, vol. 6410, pp. 232–243, 2010) showed that this problem can be solved in \(\mathcal {O}((14k +14)^{k+1} kn^{6})\) time. We improve this result by presenting an \(\mathcal {O}(6^{k} kn^{6})\) time algorithm for Proper Interval Vertex Deletion. Our fixed-parameter algorithm is based on a new structural result stating that every connected component of a {claw,net,tent,C 4,C 5,C 6}-free graph is a proper circular arc graph, combined with a simple greedy algorithm that solves Proper Interval Vertex Deletion on {claw,net,tent,C 4,C 5,C 6}-free graphs in \(\mathcal {O}(n+m)\) time. Our approach also yields a polynomial-time 6-approximation algorithm for the optimization variant of Proper Interval Vertex Deletion.

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Acknowledgements

We would like to thank Fedor V. Fomin for comments and suggestions on the manuscript and two anonymous referees for excellent reports, which helped us to significantly improve the presentation of the paper. This work has been supported by the Norwegian Research Council.

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  1. Department of Informatics, University of Bergen, 5020, Bergen, Norway

    Pim van ’t Hof & Yngve Villanger

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  1. Pim van ’t Hof
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  2. Yngve Villanger
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Correspondence to Yngve Villanger.

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van ’t Hof, P., Villanger, Y. Proper Interval Vertex Deletion. Algorithmica 65, 845–867 (2013). https://doi.org/10.1007/s00453-012-9661-3

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