Work

On the weak k-metric dimension of Hamming graphs

Year: 2026

Type: article

Abstract: Given a connected graph G , a set of vertices X ⊂ V ( G ) is a weak k -resolving set of G if for each two vertices y , z ∈ V ( G ) , the sum of the values | d G ( y , x ) − d G ( z , x ) | over all x ∈ 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 wdim k ( G ) . In this paper, wdim k ( K n □ K n ) is determined for every n ≥ 3 and every 2 ≤ k ≤ 2 n . An improvement of a known integer linear programming formulation for this problem is developed and implemented for the graphs K n □ K m . Conjectures regarding these general situations are posed.

Source: Discrete Optimization

DOI: 10.1016/j.disopt.2026.100945

Authors: Elena Fernández, Sandi Klavžar, Dorota Kuziak, Manuel Muñoz-Márquez, Ismael G. Yero

Institutions: Universidad de Cádiz, University of Ljubljana, University of Maribor, Institute of Mathematics, Physics, and Mechanics

Language: English

Topic: Graph Labeling and Dimension Problems

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,760