Graph Theory

  1. Matchings
    1. Basic Matching Concepts
      1. Matching Definition
        1. Independent Edge Sets
          1. Matched and Unmatched Vertices
          2. Types of Matchings
            1. Maximum Matching
              1. Maximal Matching
                1. Perfect Matching
                  1. Near-Perfect Matching
                  2. Matching Number
                    1. Size of Maximum Matching
                      1. Bounds and Estimates
                    2. Matchings in Bipartite Graphs
                      1. Bipartite Matching Problem
                        1. Left and Right Vertex Sets
                          1. Matching Constraints
                          2. Augmenting Paths
                            1. Alternating Paths
                              1. Augmenting Path Theorem
                              2. Hall's Marriage Theorem
                                1. Statement and Proof
                                  1. Necessary and Sufficient Conditions
                                  2. König's Theorem
                                    1. Vertex Cover Connection
                                      1. Min-Max Relationship
                                        1. Proof Techniques
                                        2. Algorithms for Bipartite Matching
                                          1. Ford-Fulkerson Reduction
                                            1. Hopcroft-Karp Algorithm
                                              1. BFS-Based Approach
                                                1. Complexity Improvements
                                            2. Matchings in General Graphs
                                              1. Challenges Beyond Bipartite Graphs
                                                1. Odd Cycles
                                                  1. Blossom Structure
                                                  2. Tutte's Theorem
                                                    1. Barrier Sets
                                                      1. Deficiency Formula
                                                        1. Characterization of Perfect Matchings
                                                        2. Edmonds' Blossom Algorithm
                                                          1. Blossom Contraction
                                                            1. Implementation Complexity
                                                          2. Weighted Matching
                                                            1. Maximum Weight Matching
                                                              1. Objective Function
                                                                1. Optimality Conditions
                                                                2. Assignment Problem
                                                                  1. Bipartite Weight Matching
                                                                    1. Cost Matrix Representation
                                                                    2. Hungarian Algorithm
                                                                      1. Dual Problem Approach
                                                                        1. Potential Functions
                                                                          1. Complexity Analysis
                                                                          2. Applications
                                                                            1. Task Assignment
                                                                              1. Resource Allocation
                                                                                1. Pairing Problems

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                                                                            10. Planar Graphs

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                                                                            12. Advanced Topics in Graph Theory