What does this research mean for the field?

Computing vertex orbits by reducing the problem to pairwise satisfiability queries, combined with structural invariants, enables the certification of exact orbit partitions and extends the practical reach on specific hard graph families compared to existing tools like Traces. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To compute and certify exact vertex orbits in graphs by reducing the equivalence problem to pairwise Boolean satisfiability queries supported by structural graph invariants.

September 2, 2026Open Access

Certified Orbit Computation by Satisfiability: Structural Invariants and their Boundaries

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To compute and certify exact vertex orbits in graphs by reducing the equivalence problem to pairwise Boolean satisfiability queries supported by structural graph invariants.
Formulated pairwise queries where models represent explicit automorphisms between vertices and unsatisfiability proves non-equivalence, deliberately excluding symmetry breaking to preserve completeness.
Accelerated resolution using structural pruning invariants, including invariant coloring from rooted decompositions, equivalence closure, feedback vertex set refinement, and Gaussian elimination for parity constraints.
Evaluated the method across 20 benchmark graph families totaling 3033 instances, comparing runtime reach and dispersion against the state-of-the-art combinatorial solver Traces under a 1 hour and 4 GB limit.
The satisfiability framework certified exact orbit partitions for 2977 of 3033 instances (98.2%), scaling up to n = 8000.
On seven challenging families, including Cai–Fürer–Immerman and Dawar–Yeung graphs, the method surpassed Traces's practical size limits by factors of 1.3 to 3.6 in n, avoiding the 174 timeouts Traces experienced among 1633 instances.
Completed runtimes within single instance sizes exhibited a dispersion of at most 14.8×, compared to up to 26620× variation observed with Traces.

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Implication

Algorithmic study demonstrates certified vertex orbit computation across 3033 benchmark graph instances, indicating greater reach and consistency on difficult graph families than combinatorial tools.

Cite This Study

Christos Karatzas (2026) studied this question.

synapsesocial.com/papers/6a97e318c562ede874ec7b84 https://doi.org/10.5281/zenodo.22214525

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Implication

Algorithmic study demonstrates certified vertex orbit computation across 3033 benchmark graph instances, indicating greater reach and consistency on difficult graph families than combinatorial tools.

Certified Orbit Computation by Satisfiability: Structural Invariants and their Boundaries

Key Points

Discussion

Implication

Cite This Study

Certified Orbit Computation by Satisfiability: Structural Invariants and their Boundaries

Key Points

Discussion

Implication

Cite This Study

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