Parameterized Complexity of Independent Set in H-Free Graphs

Abstract

In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains NP-hard for most graphs H, we study its fixed-parameter tractability and make progress towards a dichotomy between FPT and W[1]-hard cases. We first show that MIS remains W[1]-hard in graphs forbidding simultaneously \(K_{1, 4}\), any finite set of cycles of length at least 4, and any finite set of trees with at least two branching vertices. In particular, this answers an open question of Dabrowski et al. concerning \(C_4\)-free graphs. Then we extend the polynomial algorithm of Alekseev when H is a disjoint union of edges to an FPT algorithm when H is a disjoint union of cliques. We also provide a framework for solving several other cases, which is a generalization of the concept of iterative expansion accompanied by the extraction of a particular structure using Ramsey’s theorem. Iterative expansion is a maximization version of the so-called iterative compression. We believe that our framework can be of independent interest for solving other similar graph problems. Finally, we present positive and negative results on the existence of polynomial (Turing) kernels for several graphs H.

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Notes

  1. 1.

    For the sake of simplicity, “MIS ” will denote the optimisation, decision and parameterized version of the problem (in the latter case, the parameter is the size of the solution), the correct use being clear from the context.

  2. 2.

    A branching vertex in a tree is a vertex of degree at least 3.

  3. 3.

    Notice that the row-compatibility (resp. column-compatibility) relation is not symmetric.

  4. 4.

    Notice that our definition of half-graph slighly differs from the usual one, in the sense that we do not put edges relying two vertices of the same index. Hence, our construction can actually be seen as the complement of a half-graph (which is consistent with the fact that usually, both parts of a half-graph are independent sets, while they are cliques in our gadgets).

  5. 5.

    Actually, even the classical complexity of MIS in the absence of long induced paths is not well understood.

  6. 6.

    Remark that in this case, the graph induced by \(C_r \cup C_{r^{\prime }}\) is the complement of a perfect matching.

  7. 7.

    A set of vertices M is a module if every vertex \(v \notin M\) is adjacent to either all vertices of M, or none.

  8. 8.

    The non-degree of a vertex is the size of its non-neighborhood.

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Acknowledgements

N. B. and P. C. are supported by the ANR Project DISTANCIA (ANR-17-CE40-0015) operated by the French National Research Agency (ANR). We would like to thank two anonymous reviewers for their valuable comments which greatly improved the presentation of the paper.

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Correspondence to Rémi Watrigant.

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A preliminary version of this paper appeared in the proceedings of the 13th International Symposium on Parameterized and Exact Computation (IPEC 2018).

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Bonnet, É., Bousquet, N., Charbit, P. et al. Parameterized Complexity of Independent Set in H-Free Graphs. Algorithmica (2020). https://doi.org/10.1007/s00453-020-00730-6

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Keywords

  • Parameterized algorithms
  • Independent set
  • H-Free graphs