A graph-theoretic criterion for absolute irreducibility of integer-valued polynomials with square-free denominator

Abstract: An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring $\text{Int}(D)={f\in K[x]\mid f(D)\subseteq D}$, of integer-valued polynomials on a principal ideal domain $D$ with quotient field $K$, we give an easy to verify graph-theoretic sufficient condition for an element to be absolutely irreducible and show a partial converse: the condition is necessary and sufficient for polynomials with square-free denominator.