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2D Minimal Graph Rigidity is in NC for One-Crossing-Minor-Free Graphs

Authors Rohit Gurjar, Kilian Rothmund, Thomas Thierauf

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LIPIcs.STACS.2026.49.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Parallel algorithms
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Graph Rigidity
  • Parallel Algorithms
  • Polynomial Identity Testing
  • Derandomization

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Abstract

Minimally rigid graphs can be decided and embedded in the plane efficiently, i.e. in polynomial time. There is also an efficient randomized parallel algorithm, i.e. in RNC. We present an NC-algorithm to decide whether one-crossing-minor-free graphs are minimally rigid. In the special case of K_{3,3}-free graphs, we also compute an infinitesimally rigid embedding in NC.

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Rohit Gurjar, Kilian Rothmund, and Thomas Thierauf. 2D Minimal Graph Rigidity is in NC for One-Crossing-Minor-Free Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 49:1-49:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.49

Author Details

Rohit Gurjar
  • Indian Institute of Technology Bombay, India
Kilian Rothmund
  • Hochschule Aalen, Germany
  • Ulm University, Germany
Thomas Thierauf
  • Ulm University, Germany

Funding

  • Gurjar, Rohit: Supported by SERB MATRICS grant number MTR/2022/001009.
  • Rothmund, Kilian: Supported by DFG grant TH 472/5-2.
  • Thierauf, Thomas: Supported by DFG grant TH 472/5-2.

Acknowledgements

The authors thank the organizers of Dagstuhl Seminar 24381 on Algebraic and Analytic Methods in Computational Complexity for inviting them. Part of this research was done during the seminar. We thank Dhara Thakkar for helpful discussions.

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