On upper and lower bounds of identifying code set for soccer ball graph with application to satellite deployment

Abstract

We study a monitoring problem on the surface of the earth for significant environmental, social/political and extreme events using satellites as sensors. We assume that the surface of the earth is divided into a set of regions, where a region may be a continent, a country, or a set of neighboring countries. We also assume that, the impact of a significant event spills into neighboring regions and there will be corresponding indicators of such events. Careful deployment of sensors, utilizing Identifying Codes, can ensure that even though the number of deployed sensors is fewer than the number of regions, it may be possible to uniquely identify the region where the event has taken place. We assume that an event is confined to a region. As Earth is almost a sphere, we use a soccer ball (a sphere) as a model. From the model, we construct a Soccer Ball Graph (SBG), and show that the SBG has at least 26 sets of Identifying Codes of cardinality ten, implying that there are at least 26 different ways to deploy ten satellites to monitor the Earth. Finally, we also show that the size of the minimum Identifying Code for the SBG is at least nine.