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Parameterized Complexity Dichotomy for (r, ℓ)-Vertex Deletion

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Abstract

For two integers r, ℓ ≥ 0, a graph G = (V, E) is an (r, ℓ)-graph if V can be partitioned into r independent sets and ℓ cliques. In the parameterized (r, ℓ)-Vertex Deletion problem, given a graph G and an integer k, one has to decide whether at most k vertices can be removed from G to obtain an (r, ℓ)-graph. This problem is NP-hard if r + ℓ ≥ 1 and encompasses several relevant problems such as Vertex Cover and Odd Cycle Transversal. The parameterized complexity of (r, ℓ)-Vertex Deletion was known for all values of (r, ℓ) except for (2,1), (1,2), and (2,2). We prove that each of these three cases is FPT and, furthermore, solvable in single-exponential time, which is asymptotically optimal in terms of k. We consider as well the version of (r, ℓ)-Vertex Deletion where the set of vertices to be removed has to induce an independent set, and provide also a parameterized complexity dichotomy for this problem.

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Notes

  1. We would like to mention here that after this article appeared in arXiv:1310.6205, we learnt that Kolay and Panolan (arXiv:1504.08120, further published in [14]) obtained simultaneously and independently the same results that we present in Table 1 using very similar techniques.

  2. It is worth mentioning that if one is interested in optimizing the degree of the polynomial function n O(1) of our algorithms, we could solve directly the cases (1,2) and (2,1). In fact, this was the case in the original version of the paper, and Lemma 4 was added after a remark of one of the referees.

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Acknowledgments

We would like to thank the anonymous referees for helpful remarks that improved and simplified the presentation of the manuscript.

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Authors and Affiliations

  1. AlGCo Project Team, CNRS, LIRMM, Montpellier, France

    Julien Baste & Ignasi Sau

  2. FFP, Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil

    Luerbio Faria

  3. Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Sulamita Klein

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  1. Julien Baste
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  2. Luerbio Faria
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  3. Sulamita Klein
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  4. Ignasi Sau
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Corresponding author

Correspondence to Ignasi Sau.

Additional information

This work was partially supported by CNPq, CAPES, FAPERJ, and COFECUB.

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Baste, J., Faria, L., Klein, S. et al. Parameterized Complexity Dichotomy for (r, ℓ)-Vertex Deletion . Theory Comput Syst 61, 777–794 (2017). https://doi.org/10.1007/s00224-016-9716-y

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