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# Twin-Width of Graphs on Surfaces

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LIPIcs.MFCS.2024.66.pdf
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## Acknowledgements

The substantial part of the work presented in this article was done during the Brno-Koper research workshop on graph theory topics in computer science held in Kranjska Gora in April 2023, which all three authors have participated in. The authors would like to thank the anonymous reviewers for their numerous constructive comments, which helped to clarify and improve various aspects of the presentation.

## Cite As

Daniel Kráľ, Kristýna Pekárková, and Kenny Štorgel. Twin-Width of Graphs on Surfaces. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.66

## Abstract

Twin-width is a width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS'20, JACM'22], which has many structural and algorithmic applications. Hliněný and Jedelský [ICALP'23] showed that every planar graph has twin-width at most 8. We prove that the twin-width of every graph embeddable in a surface of Euler genus g is at most 18√{47g} + O(1), which is asymptotically best possible as it asymptotically differs from the lower bound by a constant multiplicative factor. Our proof also yields a quadratic time algorithm to find a corresponding contraction sequence. To prove the upper bound on twin-width of graphs embeddable in surfaces, we provide a stronger version of the Product Structure Theorem for graphs of Euler genus g that asserts that every such graph is a subgraph of the strong product of a path and a graph with a tree-decomposition with all bags of size at most eight with a single exceptional bag of size max{6, 32g-37}.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graphs and surfaces
##### Keywords
• twin-width
• graphs on surfaces
• fixed parameter tractability

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