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Searching for Smallest Universal Graphs and Tournaments with SAT

Authors Tianwei Zhang , Stefan Szeider

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Extremal graph theory
  • Software and its engineering → Constraint and logic languages
  • Hardware → Theorem proving and SAT solving
  • Mathematics of computing → Graph enumeration
Keywords
  • Constrained-based combinatorics
  • synthesis problems
  • symmetry breaking
  • SAT solving
  • subgraph isomorphism
  • tournament
  • directed graphs

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Abstract

A graph is induced k-universal if it contains all graphs of order k as an induced subgraph. For over half a century, the question of determining smallest k-universal graphs has been studied. A related question asks for a smallest k-universal tournament containing all tournaments of order k. 
This paper proposes and compares SAT-based methods for answering these questions exactly for small values of k. Our methods scale to values for which a generate-and-test approach isn't feasible; for instance, we show that an induced 7-universal graph has more than 16 vertices, whereas the number of all connected graphs on 16 vertices, modulo isomorphism, is a number with 23 decimal digits Our methods include static and dynamic symmetry breaking and lazy encodings, employing external subgraph isomorphism testing.

Cite As Get BibTex

Tianwei Zhang and Stefan Szeider. Searching for Smallest Universal Graphs and Tournaments with SAT. In 29th International Conference on Principles and Practice of Constraint Programming (CP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 280, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CP.2023.39

Author Details

Tianwei Zhang
  • Algorithms and Complexity Group, TU Wien, Austria
Stefan Szeider
  • Algorithms and Complexity Group, TU Wien, Austria

Funding

The project leading to this publication has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 101034440, and was supported by the Vienna Science and Technology Fund (WWTF) within the project ICT19-065.

Acknowledgements

This work was carried out in part while the second author visited the Simons Institute for the Theory of Computing, University of Berkeley, within the program Extended Reunion: Satisfiability. The authors thank Ciaran McCreesh for advising them to use and modify the Glasgow subgraph solver.

Supplementary Materials

References

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