Graph Theory

  1. Planar Graphs
    1. Planarity Concepts
      1. Planar Embeddings
        1. Drawing in the Plane
          1. Crossing-Free Representations
            1. Multiple Embeddings
            2. Faces of Planar Graphs
              1. Bounded and Unbounded Faces
                1. Face Boundaries
                  1. Facial Walks
                2. Euler's Formula
                  1. Statement (V - E + F = 2)
                    1. Proof for Connected Planar Graphs
                      1. Extensions to Disconnected Graphs
                        1. Corollaries and Applications
                          1. Upper Bounds on Edges
                            1. Planarity Necessary Conditions
                              1. Average Degree Bounds
                            2. Characterization of Planar Graphs
                              1. Kuratowski's Theorem
                                1. Statement
                                  1. Forbidden Subgraphs (K₅ and K₃,₃)
                                    1. Graph Subdivisions
                                      1. Homeomorphic Graphs
                                      2. Wagner's Theorem
                                        1. Graph Minors
                                          1. Minor-Closed Properties
                                        2. Planarity Testing
                                          1. Algorithmic Approaches
                                            1. Linear Time Algorithms
                                              1. Hopcroft-Tarjan Algorithm
                                              2. Complexity Analysis
                                                1. Implementation Considerations
                                                2. Planar Graph Properties
                                                  1. Dual Graphs
                                                    1. Construction Method
                                                      1. Duality Properties
                                                        1. Self-Dual Graphs
                                                        2. Planar Graph Coloring
                                                          1. Four Color Theorem Applications
                                                            1. Five Color Theorem
                                                            2. Outerplanar Graphs
                                                              1. Definition and Properties
                                                                1. Recognition Algorithms

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                                                            9. Graph Coloring

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                                                            11. Matchings