Metric dimension of Cartesian product of stars

TL;DR

This paper determines the exact metric dimension of hub-and-spoke grid graphs, provides a linear-time algorithm for constructing minimal resolving sets, and discusses implications for network localization and monitoring.

Contribution

It offers the first exact values for the metric dimension of Cartesian products of stars and a practical linear-time construction algorithm.

Findings

Exact metric dimension values for $K_{1,m} imes K_{1,n}$

Linear-time algorithm for minimal resolving sets

Visualizations of parameter regimes for design insights

Abstract

The metric dimension of a graph is the minimum number of landmark vertices required so that every vertex can be uniquely identified by its distances to the landmarks. This parameter captures the fundamental tradeoff between compact information encoding and unambiguous identification in networked systems. In this work, we determine exact value for the metric dimension of the Cartesian product $K_{1,m} \square K_{1,n}$, also known as hub-and-spoke grids, across all values of $m$ and $n$. In addition, we present a constructive linear-time algorithm that builds a minimum resolving set, providing both theoretical guarantees and practical feasibility. We complement our results with visualization of parameter regimes that illustrate the design space. The findings establish design rules for minimizing landmark sensors and support applications in graph-based localization, monitoring networks, and intelligent information systems. Our results extend the theory of metric dimension and contribute efficient methods of direct relevance to information science and computational graph theory.