Metric dimension of Cartesian product of stars

TL;DR

This paper determines the exact metric dimension of the Cartesian product of two stars, K_{1,m} ☐ K_{1,n}, across all m,n. The authors introduce a grid-based representation and analyze three regimes, yielding closed-form expressions and a constructive framework using an auxiliary graph H to realize the bounds. A linear-time algorithm is provided to construct minimum resolving sets, with direct implications for resource‑efficient sensor placement and graph-based localization in hub‑and‑spoke networks. The work advances metric dimension theory for Cartesian products and offers practical benchmarks and design rules for monitoring large structured networks.

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.

Figures (5)

• Figure 1: A grid representation of $K_{1,m} \square K_{1,n}.$
• Figure 2: A set of six panels illustrating configurations and adjacency relations in the graph $H(K_{1,m}\square K_{1,n}, B)$. (a) illustrates a path configuration of order 5 in a basis; (b) depicts the adjacency representation in a modified path structure; (c) shows an induced maximal path split into shorter resolving paths in a basis; (d) highlights the adjacency resolution of vertices in a modified $H(G, B)$; (e) represents resolving substructures in $H(G, B)$ to reduce basis size; and, (f) demonstrates adjacency in a specific subgraph of $H(G, B)$.
• Figure 3: A set of four figures demonstrating adjacency relations and path resolutions in the graph $H(G, B)$ to analyze the basis and resolving properties: (a) depicts the adjacency resolving set applied to the graph $H(G, B)$; (b) illustrates resolving configurations for paths of order $4$; (c) shows adjacency in modified resolving structures; and, (d) demonstrates adjacency resolution with vertices of degree at least $3$.
• Figure 4: Linear extension plot of the metric dimension $\dim(K_{1,m} \square K_{1,n})$ against $m$ for a fixed value of $n=14$. The plot shows the change in metric dimension as $m$ varies within the ranges $2 \leq m <\frac{n}{2}$ and $\frac{n}{2} \leq m \leq n$.
• Figure 5: Heatmap of $\dim(K_{1,m} \square K_{1,n})$ as a function of $m$ and $n$. The $x$-axis represents $n$, the $y$-axis represents $m$, and the colors indicate the metric dimension for each pair of $m$ and $n$.

Theorems & Definitions (45)

• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• Lemma 5
• proof
• Lemma 6