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Twin-Width of Graphs with Tree-Structured Decompositions

Authors Irene Heinrich , Simon Raßmann

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Author Details

Irene Heinrich
  • Technische Universität Darmstadt, Germany
Simon Raßmann
  • Technische Universität Darmstadt, Germany

Funding

  • Heinrich, Irene: The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (EngageS: grant agreement No. 820148).

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Irene Heinrich and Simon Raßmann. Twin-Width of Graphs with Tree-Structured Decompositions. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.IPEC.2023.25

Abstract

The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 [Édouard Bonnet et al., 2022; Édouard Bonnet et al., 2020], a mass of new results has appeared relating twin width to group theory, model theory, combinatorial optimization, and structural graph theory.
We take a detailed look at the interplay between the twin-width of a graph and the twin-width of its components under tree-structured decompositions: We prove that the twin-width of a graph is at most twice its strong tree-width, contrasting nicely with the result of [Édouard Bonnet and Hugues Déprés, 2023; Édouard Bonnet and Hugues Déprés, 2022], which states that twin-width can be exponential in tree-width. Further, we employ the fundamental concept from structural graph theory of decomposing a graph into highly connected components, in order to obtain optimal linear bounds on the twin-width of a graph given the widths of its biconnected components. For triconnected components we obtain a linear upper bound if we add red edges to the components indicating the splits which led to the components. Extending this approach to quasi-4-connectivity, we obtain a quadratic upper bound. Finally, we investigate how the adhesion of a tree decomposition influences the twin-width of the decomposed graph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • twin-width
  • quasi-4 connected components
  • strong tree-width

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References

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