On the weak $k$-metric dimension of Hamming graphs
Elena Fernandez, Sandi Klavzar, Dorota Kuziak, Manuel Muñoz-Marquez, Ismael G. Yero
TL;DR
The paper determines the weak $k$-metric dimension of the two-dimensional Hamming graph $K_n\square K_n$ for all feasible $k$ with $n\ge 3$, delivering a closed-form, piecewise formula that depends on the parity and size of $k$. It provides constructions and lower-bound arguments across cases $k=2$, $k=3$, and $4\le k\le 2n$, culminating in $\operatorname{wdim}_2(K_n\square K_n)=\lceil 4n/3\rceil$, $\operatorname{wdim}_3(K_n\square K_n)=2n$, and, for even $k$ or $k=3$, $\operatorname{wdim}_k(K_n\square K_n)=n\lceil k/2\rceil$, with the odd-$k\ge5$ case giving $n\lceil k/2\rceil-1$. The authors develop and compare ILP formulations, notably a layered model $F_{gh}$, to efficiently compute $\operatorname{wdim}_k(G)$ for $K_n\square K_m$, and their computational experiments corroborate the theoretical results and support broader conjectures for higher-dimensional Hamming graphs. The work advances understanding of weak metric dimensions in Cartesian/Hamming graphs and points to future extensions to $d$-dimensional cases and hypercube graphs.
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.