Minimal doubly resolving sets and the strong metric dimension of Hamming graphs

We consider the problem of determining the cardinality $psi(H_{2,k})$ of minimal doubly resolving sets of Hamming graphs $H_{2,k}.$ We prove that for $k geq 6$ every minimal resolving set of $H_{2,k}$ is also a doubly resolving set, and, consequently, $psi(H_{2,k})$ is equal to the metric dimension of $H_{2,k},$ which is known from the literature. Moreover, we find an explicit expression for the strong metric dimension of all Hamming graphs $H_{n,k}.$