On the Integer Domination Root Conjecture

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 $x=-4$.

6 pages, 1 figure