---
title: "Learning Minimally Rigid Graphs with High Realization Counts"
authors:
  - "Oleksandr Slyvka"
  - "Jan Rubeš"
  - "Rodrigo Alves"
  - "Jan Legerský"
doi: "10.24963/ijcai.2026/536"
url: "https://ijcai.org/proceedings/2026/536"
pdf: "https://ijcai.org/proceedings/2026/0536.pdf"
venue: "Proceedings of the Thirty-Fifth International Joint Conference on Artificial Intelligence"
year: 2026
pages: "4813-4821"
track: "Main Track"
keywords:
  - "Machine Learning: Evolutionary learning"
  - "Machine Learning: Reinforcement learning"
  - "Machine Learning: Sequence and graph learning"
---

# Learning Minimally Rigid Graphs with High Realization Counts

Oleksandr Slyvka, Jan Rubeš, Rodrigo Alves, Jan Legerský

*Proceedings of the Thirty-Fifth International Joint Conference on Artificial Intelligence*, pages 4813-4821. DOI: [10.24963/ijcai.2026/536](https://doi.org/10.24963/ijcai.2026/536)

## Abstract

For minimally rigid graphs, the same edge-length data can admit multiple realizations (up to translations and rotations). Finding graphs with exceptionally many realizations is an extremal problem in rigidity theory, but exhaustive search quickly becomes infeasible due to the super-exponential growth of the number of candidate graphs and the high cost of realization-count evaluation. We propose a reinforcement-learning approach that constructs minimally rigid graphs via 0- and 1-extensions, also known as Henneberg moves. We optimize realization-count invariants using the Deep Cross-Entropy Method with a policy parameterized by a Graph Isomorphism Network encoder and a permutation-equivariant extension-level action head. Empirically, our method matches the known optima for planar realization counts and improves the best known bounds for spherical realization counts, yielding new record graphs.