Independent sets in some classes of $$S_{i, j, k}$$ S i , j , k -free graphs

Karthick, T.

doi:10.1007/s10878-016-0096-7

Cited by 2 publications

(2 citation statements)

References 24 publications

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“…The complexity of MIS is still open in S 1,2,2 -free graphs and S 1,1,3 -free graphs. In this direction, the only positive results concern some subcases: it is polynomial-time solvable in (S 1,2,2 , S 1,1,3 , dart)-free graphs [17], (S 1,1,3 , banner)-free graphs and (S 1,1,3 , bull)-free graphs [18], where dart, banner and bull are particular graphs on five vertices.…”

Section: Introductionmentioning

confidence: 99%

Parameterized Complexity of Independent Set in H-Free Graphs

Bonnet¹,

Bousquet²,

Charbit³

et al. 2018

Preprint

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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 N P -hard for most graphs H, we study its fixed-parameter tractability and make progress towards a dichotomy between F P T 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 F P T 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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Section: Introductionmentioning

confidence: 99%

Parameterized Complexity of Independent Set in H-Free Graphs

Bonnet¹,

Bousquet²,

Charbit³

et al. 2018

Preprint

View full text Add to dashboard Cite

show abstract

“…The complexity of MIS is still open in S 1,2,2free graphs and S 1,1,3 -free graphs. In this direction, the only positive results concern some subcases: it is polynomial-time solvable in (S 1,2,2 , S 1,1,3 , dart)free graphs [18], (S 1,1,3 , banner)-free graphs and (S 1,1,3 , bull)-free graphs [19],…”

mentioning

confidence: 99%

Parameterized Complexity of Independent Set in H-Free Graphs

Bonnet¹,

Bousquet²,

Charbit³

et al. 2020

Algorithmica

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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 N Phard 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 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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Independent sets in some classes of $$S_{i, j, k}$$ S i , j , k -free graphs

Cited by 2 publications

References 24 publications

Parameterized Complexity of Independent Set in H-Free Graphs

Parameterized Complexity of Independent Set in H-Free Graphs

Parameterized Complexity of Independent Set in H-Free Graphs

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