UsefulLinks
Mathematics
Graph Theory
1. Introduction to Graph Theory
2. Fundamental Concepts and Types of Graphs
3. Representing Graphs
4. Paths, Walks, and Cycles
5. Graph Traversal
6. Trees and Forests
7. Shortest Path Algorithms
8. Network Flow
9. Graph Coloring
10. Planar Graphs
11. Matchings
12. Advanced Topics in Graph Theory
10.
Planar Graphs
10.1.
Planarity Concepts
10.1.1.
Planar Embeddings
10.1.1.1.
Drawing in the Plane
10.1.1.2.
Crossing-Free Representations
10.1.1.3.
Multiple Embeddings
10.1.2.
Faces of Planar Graphs
10.1.2.1.
Bounded and Unbounded Faces
10.1.2.2.
Face Boundaries
10.1.2.3.
Facial Walks
10.2.
Euler's Formula
10.2.1.
Statement (V - E + F = 2)
10.2.2.
Proof for Connected Planar Graphs
10.2.3.
Extensions to Disconnected Graphs
10.2.4.
Corollaries and Applications
10.2.4.1.
Upper Bounds on Edges
10.2.4.2.
Planarity Necessary Conditions
10.2.4.3.
Average Degree Bounds
10.3.
Characterization of Planar Graphs
10.3.1.
Kuratowski's Theorem
10.3.1.1.
Statement
10.3.1.2.
Forbidden Subgraphs (K₅ and K₃,₃)
10.3.1.3.
Graph Subdivisions
10.3.1.4.
Homeomorphic Graphs
10.3.2.
Wagner's Theorem
10.3.2.1.
Graph Minors
10.3.2.2.
Minor-Closed Properties
10.4.
Planarity Testing
10.4.1.
Algorithmic Approaches
10.4.1.1.
Linear Time Algorithms
10.4.1.2.
Hopcroft-Tarjan Algorithm
10.4.2.
Complexity Analysis
10.4.3.
Implementation Considerations
10.5.
Planar Graph Properties
10.5.1.
Dual Graphs
10.5.1.1.
Construction Method
10.5.1.2.
Duality Properties
10.5.1.3.
Self-Dual Graphs
10.5.2.
Planar Graph Coloring
10.5.2.1.
Four Color Theorem Applications
10.5.2.2.
Five Color Theorem
10.5.3.
Outerplanar Graphs
10.5.3.1.
Definition and Properties
10.5.3.2.
Recognition Algorithms
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9. Graph Coloring
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11. Matchings