Metric Dimension of Villarceau Grids

Abstract

The metric dimension of a graph measures how uniquely vertices may be identified using a set of landmark vertices. This concept is frequently used in the study of network architecture, location-based problems and communication. Given a graph $G$, the metric dimension, denoted as $\dim(G)$, is the minimum size of a resolving set, a subset of vertices such that for every pair of vertices in $G$, there exists a vertex in the resolving set whose shortest path distance to the two vertices is different. This subset of vertices helps to uniquely determine the location of other vertices in the graph. A basis is a resolving set with a least cardinality. Finding a basis is a problem with practical applications in network design, where it is important to efficiently locate and identify nodes based on a limited set of reference points. The Cartesian product of $P_m$ and $P_n$ is the grid network in network science. In this paper, we investigate two novel types of grids in network science: the Villarceau grid Type I and Type II. For each of these grid types, we find the precise metric dimension.

Figures (6)

• Figure 1: Villarceau torus
• Figure 2: (a) $VG^1_{4,5}$; (b) Acute lines are depicted in red, while obtuse lines are depicted in blue.
• Figure 3: $VG^2_{4,6}$
• Figure 4: (a) $VG^1_{4,9}$; (b) $VG^1_{6,9}$
• Figure 5: $VG^1_{4,20}$

Theorems & Definitions (20)

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