On Cayley representations of finite graphs over Abelian $ p$-groups

A polynomial-time algorithm is constructed that, given a graph $ \Gamma $, finds the full set of nonequivalent Cayley representations of $ \Gamma $ over the group $ D\cong C_p\times C_{p^k}$, where $ p\in \{2,3\}$ and $ k\geq 1$. This result implies that the recognition and isomorphism problems for Cayley graphs over $ D$ can be solved in polynomial time.