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

TL;DR

This paper investigates the weak $k$-metric dimension of Hamming graphs, providing exact values for certain cases, improving computational methods, and proposing conjectures for broader scenarios.

Contribution

It determines the weak $k$-metric dimension of $K_n \,\square\ K_n$ for all relevant parameters and enhances the ILP formulation for computing this metric.

Findings

Exact values of $ ext{wdim}_k(K_n \square K_n)$ for all $n\ge 3$ and $2\le k\le 2n$

Improved ILP formulation for calculating weak $k$-metric dimension

Proposed conjectures for general Hamming graph cases

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.