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Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width

Authors Benjamin Bergougnoux , Jakub Gajarský , Grzegorz Guśpiel , Petr Hliněný , Filip Pokrývka , Marek Sokołowski

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Fixed parameter tractability
Keywords
  • twin-width
  • tree-width
  • excluded grid
  • sparsity

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Abstract

Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS 2020]. Very briefly, its essence is a gradual reduction (a contraction sequence) of the given graph down to a single vertex while maintaining limited difference of neighbourhoods of the vertices, and it can be seen as widely generalizing several other traditional structural parameters. Having such a sequence at hand allows to solve many otherwise hard problems efficiently. Our paper focuses on a comparison of twin-width to the more traditional tree-width on sparse graphs. Namely, we prove that if a graph G of twin-width at most 2 contains no K_{t,t} subgraph for some integer t, then the tree-width of G is bounded by a polynomial function of t. As a consequence, for any sparse graph class C we obtain a polynomial time algorithm which for any input graph G ∈ C either outputs a contraction sequence of width at most c (where c depends only on C), or correctly outputs that G has twin-width more than 2. On the other hand, we present an easy example of a graph class of twin-width 3 with unbounded tree-width, showing that our result cannot be extended to higher values of twin-width.

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Benjamin Bergougnoux, Jakub Gajarský, Grzegorz Guśpiel, Petr Hliněný, Filip Pokrývka, and Marek Sokołowski. Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.11

Author Details

Benjamin Bergougnoux
  • University of Warsaw, Poland
Jakub Gajarský
  • University of Warsaw, Poland
Grzegorz Guśpiel
  • Masaryk University, Brno, Czech Republic
Petr Hliněný
  • Masaryk University, Brno, Czech Republic
Filip Pokrývka
  • Masaryk University, Brno, Czech Republic
Marek Sokołowski
  • University of Warsaw, Poland

Funding

J. Gajarský and M. Sokołowski have received funding from the European Research Council (ERC) (grant agreement No 948057 - BOBR).

References

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