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Almost All Graphs Are Vertex-Minor Universal

Authors Ruben Ascoli , Bryce Frederickson , Sarah Frederickson , Caleb McFarland , Logan Post

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ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Theory of computation → Quantum computation theory
Keywords
  • vertex-minors
  • random graphs
  • quantum networks
  • graph states

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Abstract

Answering a question of Claudet, we prove that the uniformly random graph G∼ 𝔾(n, 1/2) is Ω(√n)-vertex-minor universal with high probability. That is, for some constant α≈ 0.911, any graph on any α√ n specified vertices of G can be obtained as a vertex-minor of G. This has direct implications for quantum communications networks: an n-vertex k-vertex-minor universal graph corresponds to an n-qubit k-stabilizer universal graph state, which has the property that one can induce any stabilizer state on any k qubits using only local operations and classical communications. 
We further employ our methods in two other contexts. We obtain a bipartite pivot-minor version of our main result, and we use it to derive a universality statement for minors in random binary matroids. We also introduce the vertex-minor Ramsey number R_{vm}(k) to be the smallest value n such that every n-vertex graph contains an independent set of size k as a vertex-minor. Supported by our main result, we conjecture that R_{vm}(k) is polynomial in k. We prove Ω(k²) ≤ R_{vm}(k) ≤ 2^k - 1.

Cite As Get BibTex

Ruben Ascoli, Bryce Frederickson, Sarah Frederickson, Caleb McFarland, and Logan Post. Almost All Graphs Are Vertex-Minor Universal. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.38

Author Details

Ruben Ascoli
  • Department of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
Bryce Frederickson
  • Department of Mathematics, Emory University, Atlanta, GA, USA
Sarah Frederickson
  • Department of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
Caleb McFarland
  • Department of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
Logan Post
  • Department of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA

Funding

  • McFarland, Caleb: Supported in part by the National Science Foundation under Grant No. DMS- 2452111.

Acknowledgements

This work began at the Fall 2025 Atlanta Graduate Combinatorics Workshop at Georgia Tech. We thank Georgia Tech for providing funding for the workshop, Sean Longbrake for helping to organize it, and Elliot Frederickson for his motivating presence there. We also extend our thanks to Xiaoyu He and Rose McCarty for enlightening conversations, and to Rose McCarty for telling us about the problem of whether the uniformly random graph is vertex-minor universal (which was originally posed by Nathan Claudet) and pointing us to relevant sources. Finally, we thank Nathan Claudet for helpful comments on an earlier version of this paper.

References

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