What was already known
The conjecture allowed only 0 and −2 as integer domination roots. It held for trees and unicyclic graphs and had survived small-order searches.
AI involvement
Alikhani and Griswold credit Claude Fable 5 with the S-unit branch-cancellation mechanism. They supplied the graph, proof and exact computational validation.
The disclosed model-origin S-unit cancellation is the central constructive mechanism, while the authors present and validate the concrete graph.
Importance
The finite graph refutes the general integer-root conjecture.
Limitations
The authors do not claim 33 vertices is minimal. The independent review checked printed factorization consistency and the −4 zero, but did not recompute the polynomial from the graph’s edge list.
Technical verification
The paper reports exact set enumeration of dominating sets, an exact polynomial and rational factorization, independent author computation, and author verification of the formal arguments. A local symbolic check reproduced the printed factorization and −4 root; graph enumeration remains source-reported.
Formal checking
Exact set enumeration of dominating sets and rational polynomial factorization
- Checker
- Alikhani and Griswold's exact computation
- Scope
- The stated graph G_33 has the listed domination polynomial and x+4 factor
- Basis
- Reported by the linked checker
Authors report verification of all formal proofs in the manuscript
- Checker
- Saeid Alikhani and Max Griswold
- Scope
- G_33 counterexample and S-unit branch-cancellation argument
- Basis
- Reported by the linked checker
Symbolic expansion and integer evaluation with SymPy
- Checker
- AIProofWatch review calculation
- Scope
- The printed factorization has degree 33, gives D(-4)=0 and matches the printed d_11, d_19 and d_33 coefficients
- Basis
- Reproduced by AIProofWatch
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The authors explain the branch ratios and their odd-prime cancellation in Section 3, including how the product -8 enforces D(G_33,-4)=0.
Correction and publication history
The finite graph refutes the general integer-root conjecture.
Read this published revision