The Cardinality of Identifying Code Sets for Soccer Ball Graph with Application to Remote Sensing

TL;DR

This paper analyzes the minimum number of sensors needed for unique Earth region identification using a soccer ball graph model, providing both analytical and formal proofs for the cardinality of identifying code sets.

Contribution

It introduces a novel soccer ball graph model for satellite deployment and offers rigorous proofs for the minimal sensor sets required for unique Earth monitoring.

Findings

At least 26 ICSes of size ten exist for the SBG.

Minimum identifying code set size for the SBG is exactly ten.

At least nine sensors are needed, but the exact minimum is ten.

Abstract

In the context of satellite monitoring of the earth, we can assume that the surface of the earth is divided into a set of regions. We assume that the impact of a big social/environmental event spills into neighboring regions. Using Identifying Code Sets (ICSes), we can deploy sensors in such a way that the region in which an event takes place can be uniquely identified, even with fewer sensors than regions. As Earth is almost a sphere, we use a soccer ball as a model. We construct a Soccer Ball Graph (SBG), and provide human-oriented, analytical proofs that 1) the SBG has at least 26 ICSes of cardinality ten, implying that there are at least 26 different ways to deploy ten satellites to monitor the Earth and 2) that the cardinality of the minimum Identifying Code Set (MICS) for the SBG is at least nine. We then provide a machine-oriented formal proof that the cardinality of the MICS for the SBG is in fact ten, meaning that one must deploy at least ten satellites to monitor the Earth in the SBG model. We also provide machine-oriented proof that there are exactly 26 ICSes of cardinality ten for the SBG.