Graph Theory

  1. Network Flow
    1. Flow Networks
      1. Network Structure
        1. Directed Graph Foundation
          1. Source and Sink Vertices
            1. Capacity Function
              1. Flow Function
              2. Flow Properties
                1. Capacity Constraints
                  1. Flow Conservation
                    1. Skew Symmetry
                    2. Residual Networks
                      1. Residual Capacity
                        1. Augmenting Paths
                      2. Maximum Flow Problem
                        1. Problem Definition
                          1. Flow Value
                            1. Feasible Flows
                            2. Cuts in Flow Networks
                              1. Definition of Cuts
                                1. Cut Capacity
                                  1. Minimum Cuts
                                  2. Max-Flow Min-Cut Theorem
                                    1. Statement
                                      1. Proof Outline
                                        1. Implications
                                      2. Maximum Flow Algorithms
                                        1. Ford-Fulkerson Method
                                          1. Augmenting Path Approach
                                            1. Residual Graph Construction
                                              1. Termination Conditions
                                                1. Complexity Analysis
                                                2. Edmonds-Karp Algorithm
                                                  1. BFS for Shortest Augmenting Paths
                                                    1. Polynomial Time Complexity
                                                      1. Implementation Details
                                                      2. Push-Relabel Algorithms
                                                        1. Preflow Concept
                                                          1. Height Functions
                                                            1. Generic Algorithm
                                                          2. Applications of Network Flow
                                                            1. Maximum Bipartite Matching
                                                              1. Reduction to Max Flow
                                                                1. Construction Techniques
                                                                2. Edge-Disjoint Paths
                                                                  1. Path Finding Problems
                                                                  2. Vertex Connectivity
                                                                    1. Menger's Theorem Applications
                                                                    2. Project Selection
                                                                      1. Profit Maximization
                                                                      2. Image Segmentation
                                                                        1. Computer Vision Applications

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                                                                    7. Shortest Path Algorithms

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                                                                    9. Graph Coloring