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Computing Twin-Width via Treedepth and Vertex Integrity

Authors Robert Ganian , Mathis Rocton

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LIPIcs.STACS.2026.42.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • twin-width
  • fixed-parameter algorithms
  • treedepth
  • vertex integrity

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Abstract

Twin-width is a graph parameter that has become central to explaining the fixed-parameter tractability of first-order model checking across many graph classes. Despite its algorithmic importance, computing twin-width remains poorly understood: even recognizing graphs of twin-width at most four is NP-hard, and no fixed-parameter approximations parameterized by twin-width itself are known. A recent approach towards breaking this barrier focuses on first developing fixed-parameter algorithms for computing or approximating twin-width under parameterizations distinct from twin-width.
Our first result establishes that approximating twin-width is fixed-parameter tractable when parameterized by treedepth, thereby breaking the long-standing barrier that all previous tractable parameterizations were based on deletion distance. The proof proceeds via oriented twin-width, yielding the first constructive evidence that this variant may be easier to handle algorithmically. As our second main result, we show that computing twin-width exactly is fixed-parameter tractable with respect to vertex integrity. This constitutes the first non-trivial parameterized algorithm for computing optimal contraction sequences.

Cite As Get BibTex

Robert Ganian and Mathis Rocton. Computing Twin-Width via Treedepth and Vertex Integrity. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.42

Author Details

Robert Ganian
  • Algorithms and Complexity Group, TU Wien, Austria
Mathis Rocton
  • Algorithms and Complexity Group, TU Wien, Austria

Funding

  • Ganian, Robert: Robert Ganian acknowledges support by the FWF and WWTF Science Funds (FWF projects 10.55776/Y1329 and 10.55776/COE12, WWTF project ICT22-029).
  • Rocton, Mathis: Mathis Rocton acknowledges support by the European Union’s Horizon 2020 research and innovation COFUND programme (LogiCS@TUWien, grant agreement No 101034440), and the FWF Science Fund (FWF project Y1329).

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