Here are 100 chapter titles on Graph Theory, progressing from beginner to advanced, tailored for competitive programming:
I. Foundational Concepts (20 Chapters)
- Introduction to Graph Theory
- Types of Graphs: Directed, Undirected, Weighted, Unweighted
- Graph Representations: Adjacency Matrix, Adjacency List
- Implementing Graph Representations in C++/Java/Python
- Basic Graph Terminology: Vertices, Edges, Degrees, Paths, Cycles
- Graph Traversal: Breadth-First Search (BFS)
- Graph Traversal: Depth-First Search (DFS)
- Applications of BFS and DFS
- Connected Components and their Identification
- Cycle Detection in Graphs
- Bipartite Graph Checking
- Introduction to Graph Problems in Competitive Programming
- Representing Graphs in Competitive Programming
- Input/Output Formats for Graph Problems
- Basic Graph Algorithms and their Time Complexity
- Graph Visualization Techniques
- Practice Problems: Basic Graph Traversal
- Practice Problems: Connected Components and Cycle Detection
- Practice Problems: Bipartite Graph Checking
- Practice Problems: Graph Representation
II. Intermediate Techniques (30 Chapters)
- Shortest Path Algorithms: Dijkstra's Algorithm
- Dijkstra's Algorithm with Priority Queue
- Bellman-Ford Algorithm
- Floyd-Warshall Algorithm (All-Pairs Shortest Paths)
- Minimum Spanning Trees: Kruskal's Algorithm
- Minimum Spanning Trees: Prim's Algorithm
- Disjoint Set Union (DSU) and its Applications
- Detecting Cycles using DSU
- Topological Sort
- Strongly Connected Components (SCCs)
- Kosaraju's Algorithm for SCCs
- Articulation Points and Bridges
- Finding Articulation Points and Bridges
- Eulerian Paths and Circuits
- Hamiltonian Paths and Cycles (Introduction)
- Practice Problems: Shortest Paths
- Practice Problems: Minimum Spanning Trees
- Practice Problems: Topological Sort and SCCs
- Practice Problems: Articulation Points and Bridges
- Practice Problems: Eulerian Paths and Circuits
III. Advanced Concepts and Applications (30 Chapters)
- Advanced Graph Algorithms and Techniques
- Network Flow: Ford-Fulkerson Algorithm
- Max Flow Min Cut Theorem
- Edmonds-Karp Algorithm
- Dinic's Algorithm
- Minimum Cost Maximum Flow
- Matching in Bipartite Graphs
- Maximum Matching Algorithms
- Stable Marriage Problem
- Graph Coloring
- Chromatic Number of a Graph
- Graph Isomorphism (Basic Concepts)
- Planar Graphs and their Properties
- Geometric Graphs
- Shortest Path in a Grid (Variations)
- Traveling Salesperson Problem (TSP) (Introduction and Heuristics)
- Dynamic Programming on Graphs
- Graph Problems and Bitmasking
- Graph Problems and Number Theory
- Case Study: Solving Real-World Problems with Graph Algorithms
IV. Expert Level and Competitive Programming Challenges (20 Chapters)
- Advanced Graph Problem Sets (Codeforces, LeetCode, etc.)
- Hard Level Graph Problems and Solutions
- Contests and Challenges: Graph Theory Focus
- Analyzing Time and Space Complexity of Advanced Graph Algorithms
- Advanced Optimization Techniques for Graph Problems
- Parallel Processing with Graph Algorithms
- Distributed Graph Processing (Brief Overview)
- Implementing Graph Algorithms in Different Programming Paradigms
- Performance Tuning of Graph Implementations
- Advanced Debugging and Profiling of Graph Code
- Code Review and Best Practices for Graph Implementations
- Graph Algorithms and System Design (Rarely Applicable Directly)
- Research Topics in Graph Theory
- The Future of Graph Algorithms
- Graph Neural Networks (GNNs) (Introduction)
- Graph Databases (Brief Overview)
- Mastering Graph Theory for Competitive Programming Success
- Connecting Graph Theory to Other Algorithmic Domains
- Exploring Variations of Classic Graph Problems
- Applying Graph Theory to Real-World Scenarios (Beyond Basic Examples)
- Directed Acyclic Graphs (DAGs) and their properties
- Transitive Closure of a Graph
- Lowest Common Ancestor (LCA) in a Tree
- Tree Isomorphism
- Centroid Decomposition of a Tree
- Heavy Light Decomposition (HLD)
- Suffix Tree and Suffix Array (Brief introduction and relation to string graphs)
- String Graphs and their applications
- Planar Graph Embedding
- Graph Drawing Algorithms
- Approximation Algorithms for NP-hard Graph Problems
- Parameterized Complexity of Graph Problems
- Randomized Algorithms for Graph Problems
- Online Graph Algorithms
- Dynamic Graph Algorithms
- External Memory Graph Algorithms
- Graph Visualization Libraries and Tools
- Parallel Graph Processing Frameworks
- Distributed Graph Processing Frameworks
- Graph Theory and its Applications in various domains