  A pair of graphs $(\Gamma,\Sigma)$ is said to be stable if the full
automorphism group of $\Gamma\times\Sigma$ is isomorphic to the product of the
full automorphism groups of $\Gamma$ and $\Sigma$ and unstable otherwise, where
$\Gamma\times\Sigma$ is the direct product of $\Gamma$ and $\Sigma$. In this
paper, we reduce the study of the stability of any pair of regular graphs
$(\Gamma,\Sigma)$ with coprime valencies and vertex-transitive $\Sigma$ to that
of $(\Gamma,K_2)$. Since the latter is well studied in the literature, this
enables us to determine the stability of any pair of regular graphs
$(\Gamma,\Sigma)$ with coprime valencies in the case when $\Sigma$ is
vertex-transitve and the stability of $(\Gamma,K_2)$ is known.
