The weak $k$-metric dimension of the direct product of complete graphs

TL;DR

This paper investigates the weak k-metric dimension of the direct product of complete graphs, providing exact values for most cases and bounds for the remaining ones.

Contribution

It computes the weak k-metric dimension for the direct product of isomorphic complete graphs, advancing understanding of this graph parameter.

Findings

Exact values obtained for most graph products

Bounds established for remaining cases

Enhanced understanding of weak k-metric dimension in graph products

Abstract

The weak $k$-metric dimension of a graph is roughly understood as the cardinality of a smallest set of vertices $S$ of the graph with the property of uniquely recognizing all the vertices of the graph throughout summations of differences of distances to the vertices of $S$. The weak $k$-metric dimension of the direct product of two isomorphic complete graphs is considered in this work. Specifically, the value of such parameter is computed for almost all possibilities of these products and a bound is provided in the remaining case.