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arXiv · 2608.00109

On the Integer Domination Root Conjecture

Saeid Alikhani·Max Griswold
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Abstract

The domination integer root conjecture asserted that $0$ and $-2$ are the only integer roots of the domination polynomial $D(G, x)$ for any graph $G$. In this paper, we document a counterexample of order $n = 33$ possessing an integer domination root at $x = -4$. We provide the complete structural description of the graph $G_{33}$, present its exact domination polynomial $D(G_{33}, x)$, and demonstrate its exact rational factorization. Furthermore, we outline the structural gadget mechanism involving transfer matrices and $S$-unit branch cancellations that gives rise to non-trivial zero evaluation at $x = -4$.

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Saeid Alikhani, Max Griswold. 2026-07-31. On the Integer Domination Root Conjecture. https://arxiv.org/abs/2608.00109

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