The connected dominating set (CDS) has been a classic subject studied in graph theory since 1975. It has been discovered in recent years that CDS has important applications in communication networks -especially in wireless networks -as a virtual backbone. Motivated from those applications, many papers have been published in the literature during last 15 years. Now, the connected dominating set has become a hot research topic in computer science. This work is a valuable reference for researchers in computer science and operations research, especially in areas of theoretical computer science, computer communication networks, combinatorial optimization, industrial engineering, and discrete mathematics. The book may also be used as a text in a graduate seminar for PhD students. Readers should have a basic knowledge of computational complexity and combinatorial optimization. In this book, the authors present the state-of-the-art in the study of connected dominating sets. Each chapter is devoted to one problem, and consists of three parts: motivation and overview, problem complexity analysis, and approximation algorithm designs. The text is designed to give the reader a clear understanding of the background, formulation, existing important research results, and open problems. Topics include minimum CDS, routing-cost constrained CDS, weighted CDS, directed CDS, SCDS (strongly connected dominating set), WCDS (weakly connected dominating set), CDS-partition, virtual backbone in wireless networks, convertor placement in optical networks, coverage in wireless sensor networks, and more.