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arXiv · 2505.19642

On the weak $k$-metric dimension of Hamming graphs

Elena Fernandez·Sandi Klavzar·Dorota Kuziak·Manuel Muñoz-Marquez·Ismael G. Yero
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Abstract

Given a connected graph $G$, a set of vertices $X\subset V(G)$ is a weak $k$-resolving set of $G$ if for each two vertices $y,z\in V(G)$, the sum of the values $|d_G(y,x)-d_G(z,x)|$ over all $x\in X$ is at least $k$, where $d_G(u,v)$ stands for the length of a shortest path between $u$ and $v$. The cardinality of a smallest weak $k$-resolving set of $G$ is the weak $k$-metric dimension of $G$, and is denoted by $\mathrm{wdim}_k(G)$. In this paper, $\mathrm{wdim}_k(K_n\,\square\,K_n)$ is determined for every $n\ge 3$ and every $2\le k\le 2n$. An improvement of a known integer linear programming formulation for this problem is developed and implemented for the graphs $K_n\,\square\,K_m$. Conjectures regarding these general situations are posed.

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Elena Fernandez, Sandi Klavzar, Dorota Kuziak, Manuel Muñoz-Marquez, Ismael G. Yero. 2025-05-26. On the weak $k$-metric dimension of Hamming graphs. https://arxiv.org/abs/2505.19642

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