In an era where network reliability is paramount, this work introduces a novel approach to fault-tolerant system design through the lens of graph theory. The book presents the innovative concept of "closely-connected vertices," a powerful framework that revolutionizes our understanding of network resilience and fault tolerance. Through meticulous analysis of cc-domination, graph polynomials, and entropy measures, readers will discover new methodologies for designing robust networks that minimize link failures. This comprehensive study bridges theoretical mathematics and practical...