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Graphs with No Long Claws: An Improved Bound for the Analog of the Gyárfás' Path Argument

Authors Romain Bourneuf , Jana Masaříková , Wojciech Nadara , Marcin Pilipczuk

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  • Mathematics of computing → Graph algorithms
Keywords
  • long-claw-free graphs
  • extended strip decomposition
  • maximum weight independent set
  • Gyárfás' path
  • three in a tree

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Abstract

For a fixed integer t ⩾ 1, a (t-)long claw, denoted S_{t,t,t}, is the unique tree with three leaves, each at distance exactly t from the vertex of degree three. Majewski et al. [ICALP 2022, ACM ToCT 2024] proved an analog of the Gyárfás' path argument for S_{t,t,t}-free graphs: given an n-vertex S_{t,t,t}-free graph, one can delete neighborhoods of 𝒪(log n) vertices so that the remainder admits an extended strip decomposition (an appropriate generalization of partition into connected components) into particles of multiplicatively smaller size. In this work, we refine the argument of Majewski et al. to its arguably final form: we show that a constant number of neighborhoods suffice. 
The statement of Majewski et al. is one of the two pillars of a recent quasi-polynomial time algorithm for Maximum Weight Independent Set in S_{t,t,t}-free graphs [Gartland et al., STOC 2024]; our work immediately improves the quasi-polynomial function in the running time bound. Furthermore, our result significantly simplifies known polynomial-time algorithms for Maximum Weight Independent Set in S_{t,t,t}-free graphs with an additional sparsity assumption such as bounded degree or excluding a fixed biclique as a subgraph.

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Romain Bourneuf, Jana Masaříková, Wojciech Nadara, and Marcin Pilipczuk. Graphs with No Long Claws: An Improved Bound for the Analog of the Gyárfás' Path Argument. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.28

Author Details

Romain Bourneuf
  • Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400, Talence, France
Jana Masaříková
  • University of Warsaw, Poland
Wojciech Nadara
  • University of Warsaw, Poland
  • Technical University of Denmark, Kongens Lyngby, Denmark
Marcin Pilipczuk
  • University of Warsaw, Poland

Funding

  • Masaříková, Jana: Supported by Polish National Science Centre PRELUDIUM 21 grant number 2022/45/N/ST6/04232.
  • Nadara, Wojciech: Supported by Independent Research Fund Denmark grant 2020-2023 (9131-00044B) "Dynamic Network Analysis" (while being employed in Denmark) and European Union’s Horizon 2020 research and innovation programme, grant agreement No. 948057 - BOBR (while being employed in Poland).
  • Pilipczuk, Marcin: Supported by Polish National Science Centre SONATA BIS-12 grant number 2022/46/E/ST6/00143.

Acknowledgements

The majority of this work was done during the Sparse Graphs Coalition workshop "Algebraic, extremal, and structural methods and problems in graph colouring" that happened online in February 2024.

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