The order-33 integer domination-root minimality claim
Abstract
A connected 21-vertex ordinary-domination witness at minus four refutes the complete order-33 minimality claim.
Alikhani and Griswold, arXiv:2608.00109v1, Section 4 item 2 asks whether n=33 is the minimal order of a finite simple connected graph whose ordinary domination polynomial vanishes at -4. The formal claim retains both attainment at order 33 and nonvanishing at every positive order below 33.
Definition 1.1 (The complete order-33 minimality claim).
Formalization. D5/S3/Combinatorics/Graph/Domination/AlikhaniGriswoldMinOrderRefutation.claim (✓ std3).
Citation. Saeid Alikhani; Max Griswold (2026). On the Integer Domination Root Conjecture. DOI: 10.48550/arXiv.2608.00109. URL: https://arxiv.org/abs/2608.00109v1.
Commentary.
The first conjunct asserts an order-33 connected graph with ordinary domination evaluation zero at -4. The second conjunct asserts that every positive n below 33 and every connected graph on Fin n has nonzero ordinary domination evaluation. Dominating subsets are counted once and need not be connected.
Theorem 1.2 (A 21-vertex counterexample refutes minimality).
Proof. Machine-checked in Lean as D5/S3/Combinatorics/Graph/Domination/AlikhaniGriswoldMinOrderRefutation.result (✓ std3). ∎
Resolves. Problems/alikhani-griswold-2026-minus-four-minimum-order (refuted) by D5/S3/Combinatorics/Graph/Domination/AlikhaniGriswoldMinOrderRefutation.result.
Source. Repository-derived.
Acknowledgement. Saeid Alikhani; Max Griswold (2026). On the Integer Domination Root Conjecture. DOI: 10.48550/arXiv.2608.00109. URL: https://arxiv.org/abs/2608.00109v1.
Commentary.
The explicit 21-vertex graph has the 25 listed undirected edges. Its core is induced on 0 through 6, with branches 7 through 12 and 13 through 20 rooted at 7 and 13 and incident on core sets {0} and {0,3,4}. The accepted ordinary decomposition gives message pairs (-1024,972) and (-3328,6912); the same supplied decomposition and exact table cancellation yield evaluation zero at -4. The graph is connected by the displayed paths from vertex 0. Since 1 <= 21 < 33, this contradicts the lower-bound conjunct and answers the source question negatively. No claim that 21 is minimal is made.
The result is a literal refutation of the complete source claim; it does not assert that 21 is the least counterexample order.