¶ Graph Algorithms
¶ Competitive programming
Graph algorithms have a special place in competitive programming, and anyone who has spent time solving problems in this world eventually realizes that graphs are not just another topic—they’re the backbone of countless challenges. Whenever a problem involves relationships, dependencies, movement, connections, or anything that can be represented as a set of objects linked together, there’s almost always a graph hiding inside it. Learning to recognize and work with graphs is one of the biggest milestones in a competitive programmer’s journey, and this course of one hundred articles is designed to take you from the very fundamentals to some of the most sophisticated techniques used in modern contests.
It doesn’t matter whether you’re a beginner who finds the idea of nodes and edges abstract, or an intermediate coder who has implemented BFS and DFS but hasn’t yet ventured into the deeper waters of flows, bridges, articulation points, or centroid decomposition. Graph theory meets you where you are and then pushes you forward, giving you tools to think more systematically about problems. And the more familiar you become with graph algorithms, the more you start seeing them everywhere—in paths, routes, choices, transitions, and dependencies that didn’t initially look like graphs at all.
There is something incredibly intuitive yet wonderfully complex about graphs. They represent life in ways few other mathematical structures do. Cities connected by roads, computers connected by cables, social networks made of friendships, tasks linked by constraints—graphs are woven into everything. In competitive programming, they’re not just a convenient model; they’re often the only correct model. Graphs allow us to formalize vague ideas into precise structures, and once that structure is clear, the problem begins to unravel.
That clarity is what this course hopes to cultivate. Graphs can feel overwhelming because they have so many variants: directed, undirected, weighted, unweighted, cyclic, acyclic, planar, bipartite. But these distinctions exist not to confuse you—they exist to help you ask the right questions. Does direction matter? Do weights matter? Do cycles exist? Is coloring possible? Every type of graph tells you something about the nature of the problem. Once you learn to identify these characteristics quickly, you begin to solve problems faster and with more confidence.
One of the most common misconceptions about graph algorithms is that they’re all about finding the shortest path or exploring nodes. While BFS and DFS are pillars of the subject—and mastering them alone unlocks an enormous number of problems—graphs extend much further. They give rise to entire families of algorithms: minimum spanning trees, topological sorting, strongly connected components, matching, flow networks, cut sets, tree decompositions, Euler tours, LCA computations, and more. Graph algorithms don’t just improve your competitive programming skill—they broaden the very way you think about logic and connection.
What makes graph problems especially satisfying is how they blend intuition with rigor. Sometimes you solve a problem by imagining water flowing through pipes, or by picturing travelers exploring cities. Other times you think in terms of constraints and ordering, like figuring out the sequence of tasks that must precede one another. And sometimes, you work purely algebraically, manipulating adjacency lists and bitsets without ever drawing a picture. Graphs give you the freedom to approach problems visually, logically, or analytically as you prefer.
As you progress through this course, you’ll learn that even the simplest graph ideas can have profound consequences. Depth-first search, for example, is so powerful it almost feels like a magic trick once you start applying it to detect cycles, classify edges, compute subtree sizes, find bridges, determine articulation points, perform topological sorts, and even optimize DP on trees. On the other hand, breadth-first search transforms the task of finding shortest paths in unweighted graphs into something almost effortless—just level-by-level exploration. These two algorithms alone solve a huge chunk of problems, yet mastering their subtleties takes time and practice. That’s why this course treats them not as trivial topics but as gateways to deeper insights.
Graph problems also teach you something rarely discussed: the importance of representation. The way you structure your graph in memory—adjacency lists, matrices, edge lists—changes the speed, clarity, and direction of your solution. The right representation turns messy logic into clean loops; the wrong one leads to confusion or inefficiency. Learning to choose the right representation is part of what separates strong competitive programmers from those who struggle with graph-heavy problems.
Another interesting aspect is that graph algorithms are often a balance between brute force and clever pruning. At times you’ll explore every node, but other times you’ll discover that certain nodes or edges are irrelevant. You’ll learn when to search deeply and when to step back and build higher-level abstractions. You’ll learn that sometimes the graph must be decomposed, condensed, or transformed before you can work with it effectively. You’ll experience the elegance of reducing a strongly connected component to a single node, the satisfaction of simplifying a directed graph into layers, and the power of converting a complex network into a flow model that reveals hidden structure.
Competitive programming pushes you to be both meticulous and intuitive with graphs. It asks you to think like a traveler, a planner, a network engineer, and a mathematician all at once. And as you solve more graph problems, you develop an instinct for how graphs behave. You begin to foresee bottlenecks, anticipate cycles, and notice patterns that others overlook. When you read a problem, you start asking: What am I really connecting here? What is the movement? What depends on what? Where is the structure hidden?
One of the great pleasures of graph theory is discovering how interconnected topics truly are. A shortest path problem might seem unrelated to minimum spanning trees, yet they share underlying structures. Flow networks might seem specialized, yet they pop up in problems about scheduling, matching, resource allocation, sports rankings, or even assigning people to seats. Tree algorithms might feel like a narrow niche, but trees form the backbone of countless problems—from DP optimization to range queries and binary lifting. Each article in this course will build on the previous ones, helping you form a cohesive understanding rather than a collection of isolated techniques.
Something competitive programmers often grapple with is complexity. Graph problems can easily become too slow if you’re not careful. That’s why part of this course focuses on learning to think not only about correctness but also efficiency. When should you use Dijkstra instead of BFS? When is a priority queue helpful? When does a problem need 𝑂(𝑁 log N) preprocessing? And when do you absolutely need a linear-time algorithm? The ability to judge time complexity in graph problems is almost as important as designing the algorithm itself.
Perhaps what sets graph algorithms apart from many other topics is how alive they feel. When you work with graphs, you’re not just crunching numbers or optimizing formulas. You’re traversing states, exploring branches, uncovering routes, isolating components, and interpreting behavior. The algorithms almost feel like living processes—moves, decisions, flows, growth, collapse, spread. Because graphs model dynamic relationships, solving graph problems often feels like telling a story: watching connections emerge, dissolve, and evolve under the rules you define.
This course doesn’t aim to overwhelm you with formal definitions or dense mathematical abstractions. Instead, it guides you through ideas in a way that feels close to how you naturally reason. You’ll build intuition step by step, discover patterns through examples, and reinforce your understanding through increasingly challenging problems. By the end, you won’t just know graph algorithms—you’ll think in graphs.
Choosing to embark on a deep study of graph algorithms means stepping into one of the richest and most foundational areas of competitive programming. It means learning to work with structures that model everything from everyday networks to abstract mathematical constructions. It means gaining tools that help you attack problems with clarity and strategy. And it means building the kind of flexible, adaptive thinking that makes you a stronger problem solver overall.
As you journey through the hundred articles ahead, you’ll uncover techniques that solve tiny puzzles and strategies that tackle massive networks. You’ll learn about trees and cycles, flows and cuts, matchings and colorings, shortest paths and decompositions, and so much more. With each new idea, your ability to understand and build algorithms will expand. Problems that once felt intimidating will begin to feel natural. And over time, you’ll find that graph algorithms have transformed not only how you code, but how you think.
Welcome to this exploration of graph algorithms. If you enjoy seeing structure in chaos, discovering the hidden patterns in connections, and solving challenges that bring logic and imagination together, then you’re exactly where you need to be. This course will take you through one of the most profound journeys in competitive programming, and by the end of it, you’ll be able to confidently navigate the networks, relationships, and structures that define some of the hardest problems in the field.
¶ Graph Algorithms
I. Foundations (20 Chapters)
1. Introduction to Graph Theory: Definitions and Terminology
2. Representing Graphs: Adjacency Matrix vs. Adjacency List
3. Graph Traversal Algorithms: Breadth-First Search (BFS)
4. Graph Traversal Algorithms: Depth-First Search (DFS)
5. Connected Components: Finding and Counting
6. Bipartite Graphs: Checking and Coloring
7. Cycle Detection: Identifying Cycles in Graphs
8. Directed vs. Undirected Graphs: Properties and Differences
9. Weighted Graphs: Introducing Edge Weights
10. Graph Data Structures: Implementing Graphs in Code
11. Basic Graph Problems: Finding Paths, Connectivity
12. Implementing BFS and DFS: Code Examples and Optimizations
13. Applications of BFS: Shortest Paths in Unweighted Graphs
14. Applications of DFS: Topological Sort, Cycle Detection
15. Representing and Manipulating Graphs in Code
16. Introduction to Graph Visualization
17. Graph Properties: Degree, Diameter, Radius
18. Special Graphs: Trees, DAGs, Complete Graphs
19. Graph Isomorphism: Basic Concepts
20. Practice Problems: Applying Basic Graph Algorithms
II. Intermediate Techniques (25 Chapters)
21. Shortest Path Algorithms: Dijkstra's Algorithm
22. Dijkstra's Algorithm: Implementation and Optimizations
23. Shortest Path Algorithms: Bellman-Ford Algorithm
24. Bellman-Ford Algorithm: Handling Negative Weights
25. All-Pairs Shortest Paths: Floyd-Warshall Algorithm
26. Minimum Spanning Trees: Kruskal's Algorithm
27. Minimum Spanning Trees: Prim's Algorithm
28. Disjoint Set Union (DSU): Implementing and Applications
29. Topological Sort: Kahn's Algorithm and DFS Approach
30. Strongly Connected Components: Kosaraju's Algorithm
31. Bridges and Articulation Points: Finding Critical Edges/Vertices
32. Eulerian Paths and Circuits: Finding and Constructing
33. Hamiltonian Paths and Cycles: Introduction and Complexity
34. Maximum Flow Algorithms: Ford-Fulkerson Algorithm
35. Maximum Flow Algorithms: Edmonds-Karp Algorithm
36. Minimum Cut: Max-Flow Min-Cut Theorem
37. Bipartite Matching: Hopcroft-Karp Algorithm
38. Network Flow Applications: Assignment Problems
39. Graph Coloring: Introduction and Greedy Approach
40. Planar Graphs: Basic Concepts
41. Graph Isomorphism: Advanced Techniques
42. Graph Decomposition: Finding Cliques and Independent Sets
43. Tree Algorithms: Lowest Common Ancestor (LCA)
44. Tree Algorithms: Diameter of a Tree
45. Practice Problems: Applying Intermediate Graph Algorithms
III. Advanced Strategies (30 Chapters)
46. Advanced Shortest Path Algorithms: Johnson's Algorithm
47. Shortest Paths in Directed Acyclic Graphs (DAGs)
48. Dynamic Programming on Graphs: Path Counting
49. Graph Algorithms and Linear Algebra: Adjacency Matrix Powers
50. Spectral Graph Theory: Introduction
51. Graph Embeddings: Representing Graphs in Vector Space
52. Graph Neural Networks (GNNs): Introduction
53. Advanced Network Flow Algorithms: Push-Relabel
54. Minimum Cost Flow: Cycle Canceling and Successive Shortest Paths
55. Matching in General Graphs: Blossom Algorithm
56. Graph Coloring: Chromatic Number and Graph Polynomials
57. Planar Graph Algorithms: Embedding and Duality
58. Geometric Graphs: Intersection Graphs
59. Graph Algorithms for Distributed Systems
60. Parallel Graph Algorithms: Shared Memory and Distributed Memory
61. Graph Algorithms for Big Data: Scaling Graph Processing
62. Graph Databases: Introduction and Query Languages
63. Graph Visualization: Advanced Techniques
64. Graph Algorithms in Machine Learning: Graph-Based Learning
65. Graph Algorithms in Data Mining: Community Detection
66. Graph Algorithms in Social Networks: Influence Maximization
67. Graph Algorithms in Bioinformatics: Sequence Alignment
68. Graph Algorithms in Computer Vision: Image Segmentation
69. Graph Algorithms in Robotics: Path Planning
70. Graph Algorithms in Game Theory: Game Tree Search
71. Advanced Graph Decomposition Techniques: Modular Decomposition
72. Advanced Tree Algorithms: Centroid Decomposition
73. Advanced Graph Isomorphism Techniques
74. Practice Problems: Tackling Challenging Graph Problems
75. Analyzing Graph Algorithms: Correctness and Complexity
IV. Expert Level & Applications (25 Chapters)
76. Graph Algorithms in Competitive Programming Contests
77. Identifying Graph Problems in Contest Settings
78. Implementing Graph Algorithms Efficiently for Contests
79. Debugging Complex Graph Algorithms
80. Advanced Graph Algorithms: Beyond the Basics
81. Graph Algorithms and Combinatorial Optimization
82. Graph Algorithms and Approximation Algorithms
83. Graph Algorithms and Randomized Algorithms
84. Graph Algorithms and Parallel Computing
85. Graph Algorithms and Distributed Computing
86. Graph Algorithms in Real-World Systems: Case Studies
87. Graph Algorithms in Software Engineering: Dependency Analysis
88. Graph Algorithms in Hardware Design: Circuit Design
89. Graph Algorithms in Cloud Computing: Resource Allocation
90. Graph Algorithms in IoT: Network Management
91. Open Problems in Graph Algorithms: Research Directions
92. The Future of Graph Algorithms: Emerging Trends
93. Graph Algorithms in Quantum Computing
94. Graph Algorithms and Cryptography
95. Graph Algorithms and Security
96. Graph Algorithms and Optimization
97. Graph Algorithms and Simulation
98. Graph Algorithms and Modeling
99. Graph Algorithms and AI
100. The Impact of Graph Algorithms: A Retrospective