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

Abstract

We consider the problem of determining the cardinality ?(H2,k) of minimal doubly resolving sets of Hamming graphs H2,k. We prove that for k ? 6 every minimal resolving set of H2,k is also a doubly resolving set, and, consequently, ?(H2,k) is equal to the metric dimension of H2,k, which is known from the literature. Moreover, we find an explicit expression for the strong metric dimension of all Hamming graphs Hn,k.