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

Abstract

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}.

On the strong metric dimension of Cartesian and direct products of graphs

“…The strong metric dimension of Hamming graphs was obtained in [15] where the authors gave a long and complicated proof. Here we give a simple proof for this result, using Theorem 3 and the next result due to Valencia-Pabon and Vera [27].…”

Edge Version of Metric Dimension and Doubly Resolving Sets of the Necklace Graph

“…The minimum cardinality of a doubly resolving set for G is represented by ψ(G). In case of some convex prism, hamming and polytopes graphs, the minimal doubly resolving sets have been obtained in [24][25][26] respectively.…”

The strong metric dimension of generalized Sierpiński graphs with pendant vertices

“…After the publication of the first paper [16], the strong metric dimension has been extensively studied. The reader is invited to read, for instance, the following works [10,11,12,13,15] and the references cited therein. For some basic graph classes, the strong metric dimension is easy to compute.…”