Front Cover; Submodular Functions and Electrical Networks; Copyright Page; Contents; Chapter 1. Introduction; Chapter 2. Mathematical Preliminaries; 2.1 Sets; 2.2 Vectors and Matrices; 2.3 Linear Inequality Systems; 2.4 Solutions of Exercises; 2.5 Solutions of Problems; Chapter 3. Graphs; 3.1 Introduction; 3.2 Graphs: Basic Notions; 3.3 Graphs and Vector Spaces; 3.4 Basic Operations on Graphs and Vector Spaces; 3.5 Problems; 3.6 Graph Algorithms; 3.7 Duality; 3.8 Notes; 3.9 Solutions of Exercises; 3.10 Solutions of Problems; Chapter 4. Matroids; 4.1 Introduction 4.2 Axiom Systems for Matroids4.3 Dual of a Matroid; 4.4 Minors of Matroids ; 4.5 Connectedness in Matroids; 4.6 Matroids and the Greedy Algorithm; 4.7 Notes; 4.8 Solutions of Exercises; Chapter 5. Electrical Networks; 5.1 Introduction; 5.2 In Terms of Multiterminal Devices; 5.3 In Terms of 2-Terminal Devices ; 5.4 Standard Devices; 5.5 Common Methods of Analysis; 5.6 Procedures used in Circuit Simulators; 5.7 State Equations for Dynamic Networks; 5.8 Multiports in Electrical Networks; 5.9 Some Elementary Results of Network Theory; 5.10 Notes; 5.11 Solutions of Exercises Chapter 6. Topological Hybrid Analysis6.1 Introduction; 6.2 Electrical Network: A Formal Description; 6.3 Some Basic Topological Results ; 6.4 A Theorem on Topological Hybrid Analysis; 6.5 Structure of Constraints and Optimization; 6.6 Notes; 6.7 Solutions of Exercises; Chapter 7. The Implicit Duality Theorem and Its Applications; 7.1 The Vector Space Version; 7.2 *Quasi Orthogonality; 7.3 Applications of the Implicit Duality Theorem; 7.4 *Linear Inequality Systems; 7.5 *Integrality Systems; 7.6 Problems; 7.7 Notes; 7.8 Solutions of Exercises; 7.9 Solutions of Problems Chapter 8. Multiport Decomposition8.1 Introduction; 8.2 Multiport Decomposition of Vector Spaces; 8.3 Analysis through Multiport Decomposition; 8.4 Port Minimization; 8.5 *Multiport Decomposition for Network Reduction; 8.6 Problems; 8.7 Solutions of Exercises; 8.8 Solutions of Problems; Chapter 9. Submodular Functions; 9.1 Introduction; 9.2 Submodularity; 9.3 Basic Operations on Semimodular Functions; 9.4 *Other Operations on Semimodular Functions; 9.5 Polymatroid and Matroid Rank Functions; 9.6 Connectedness for Semimodular Functions; 9.7 *Semimodular Polyhedra 9.8 Symmetric Submodular functions9.9 Problems; 9.10 Notes; 9.11 Solutions of Exercises; 9.12 Solutions of Problems; Chapter 10. Convolution of Submodular Functions; 10.1 Introduction; 10.2 Convolution; 10.3 Matroids, Polymatroids and Convolution; 10.4 The Principal Partition; 10.5 *The Refined Partial Order of the Principal Partition; 10.6 Algorithms for PP; 10.7 *Aligned Polymatroid Rank Functions; 10.8 Notes; 10.9 Solutions of Exercises; 10.10 Solutions of Problems; Chapter 11. Matroid Union; 11.1 Introduction; 11.2 Submodular Functions induced through a Bipartite Graph 11.3 Matroid Union: Algorithm and Structure | 
