Here are 100 chapter titles for a book on Graph Theory Basics in competitive programming, organized from beginner to advanced topics:
- Introduction to Graph Theory in Competitive Programming
- What is a Graph? Basic Definitions and Concepts
- Types of Graphs: Directed and Undirected
- Vertices, Edges, and Their Importance
- Graph Representation: Adjacency Matrix and Adjacency List
- Understanding Degree of a Vertex
- Walks, Paths, and Cycles in Graphs
- Simple Graphs, Multi-Graphs, and Pseudographs
- Connected and Disconnected Graphs
- Subgraphs and Induced Subgraphs
- Graph Isomorphism: Basic Concept and Properties
- Understanding Bipartite Graphs
- Weighted and Unweighted Graphs
- Graph Traversals: BFS vs. DFS
- Depth-First Search (DFS) Algorithm
- Breadth-First Search (BFS) Algorithm
- Applications of BFS and DFS in Graph Theory
- Topological Sorting of Directed Acyclic Graphs (DAG)
- Understanding the Shortest Path Problem
- Graph Connectivity: Basic Definitions and Algorithms
- Strongly Connected Components (SCC) in Directed Graphs
- Finding Strongly Connected Components Using Kosaraju's Algorithm
- Finding SCC Using Tarjan’s Algorithm
- Graph Coloring: Basics and Applications
- DFS and BFS for Finding Connected Components
- Eulerian Paths and Circuits
- Hamiltonian Paths and Cycles
- Planar Graphs and Their Properties
- Graph Representations: Incidence Matrix and Edge List
- Connected vs. Disconnected Components in Graphs
- Spanning Trees: Definition and Properties
- Prim’s Algorithm for Minimum Spanning Tree
- Kruskal’s Algorithm for Minimum Spanning Tree
- Dijkstra's Algorithm for Shortest Path
- Bellman-Ford Algorithm for Shortest Path
- Floyd-Warshall Algorithm for All-Pairs Shortest Path
- Breadth-First Search for Finding Shortest Path in Unweighted Graphs
- Bidirectional BFS for Faster Path Finding
- Handling Negative Weight Edges in Graphs
- Graph Representation in Competitive Programming: Trade-offs
- Dynamic Programming on Graphs: Basics and Techniques
- Graph Traversals on Large Graphs: Optimizations
- Floyd-Warshall Algorithm and Path Reconstruction
- Shortest Path Algorithms with Constraints
- A Search Algorithm for Pathfinding*
- Handling Graph Cycles: Negative Cycles in Bellman-Ford
- Johnson’s Algorithm for All-Pairs Shortest Path
- Graph Matching Algorithms: Basic Concepts
- Maximum Bipartite Matching Using DFS
- Hungarian Algorithm for Maximum Matching
- Kuhn-Munkres Algorithm for Optimal Assignment
- Maximum Flow Problem in Graphs
- Ford-Fulkerson Algorithm for Maximum Flow
- Edmonds-Karp Algorithm for Maximum Flow
- Dinic’s Algorithm for Maximum Flow
- Min-Cost Max-Flow Problem
- Graph Cut and Max Flow Min Cut Theorem
- Network Flow Applications in Competitive Programming
- Strongly Connected Components in Directed Graphs
- Tarjan’s Algorithm for Finding Bridges in Graphs
- Finding Articulation Points in Graphs
- Biconnected Components in Graphs
- Cycle Detection in Directed Graphs
- Eulerian Circuit and Path Algorithms in Directed Graphs
- Graph Embedding and Graph Representations
- Advanced Techniques for Graph Traversals
- Graph Decomposition and Dynamic Connectivity
- Graph Partitioning Algorithms
- Greedy Algorithms for Graph Problems
- Approximation Algorithms for Graph Problems
- Planarity Testing in Graph Theory
- Ramsey Theory and Graph Coloring
- Graph Isomorphism Problem and Algorithms
- Tree Algorithms: Lowest Common Ancestor (LCA)
- Heavy-Light Decomposition for Tree Queries
- Centroid Decomposition for Trees
- Euler Tour Technique and Applications
- Heavy-Light Decomposition in Path Queries
- Dynamic Trees and Link-Cut Trees
- Finding Maximum Independent Set in Graphs
- Graph Representation with Hashing Techniques
- Minimum Spanning Forest for Disconnected Graphs
- Depth-First Search (DFS) with Path Compression
- Two-Pointer Technique in Graphs
- Euler Tour and Applications in Range Queries
- Graph Algorithms on Large Datasets
- Online Algorithms for Graph Problems
- Randomized Algorithms for Graph Problems
- Graph Algorithms in Parallel Computing
- Graph Algorithms with Large-Scale Data
- Optimizing Space Complexity in Graph Algorithms
- Graph Algorithms in Geospatial Applications
- Graph Algorithms for Social Network Analysis
- Graph Algorithms in Computational Biology
- Applications of Graph Theory in Cryptography
- Graph Algorithms for Recommendation Systems
- Quantum Computing and Graph Algorithms
- Machine Learning Applications Using Graphs
- Graph Theory and Game Theory Applications
- Future Trends in Graph Theory Algorithms and Techniques
These chapter titles cover a wide range of graph theory topics, starting with basic graph concepts and progressing through more advanced algorithms and their applications in competitive programming. Each chapter builds upon the previous one, allowing readers to develop a strong foundation in graph theory and its use in algorithmic problem-solving.