The automorphism group of a product of graphs

In a recent paper we showed that every connected graph can be written as a weak cartesian product of a family of indecomposable rooted graphs and that this decomposition is unique to within isomorphisms. Using this unique prime factorization theorem we prove that if a graph $X$ can be written as a product of connected rooted graphs, which are pairwise relatively prime, then the automorphism group of $X$ is isomorphic to the restricted direct product of the automorphism groups of the factors with prescribed subgroups the isotropy groups of the factors at the roots. This is a generalization of Sabidussi’s theorem for cartesian multiplication.