Fair Detour Domination in the Corona of Two Graphs: Optimizing CCTV Camera Installation for Efficient Surveillance

Abstract

The objective of our research is to analyze the properties of FDD sets in the corona of two graphs and explore their practical application in CCTV camera installation. A set F ⊆ V(G) is considered to be a Fair Detour Dominating set (FDD-set) if it is detour dominating and the number of neighbors within set F is the same for any pair of vertices outside of F. Among these FDD sets, the fγd-set refers to the FDD-set with the smallest number of vertices, and its order defines the Fair Detour Domination number (fγd  (G)). We have established that for any arbitrary graphs G1 and  G2, fγd (G1 o G2)   = |V(G1) iff fd(G2)  = 1 we have determined the  fγd number of corona products of any connected graph G with the path graph as well as the cycle graph. We also characterized FDD sets in the corona product of two connected graphs and provided a thorough description of how FDD sets can be used in optimizing CCTV camera installation for efficient surveillance.