Graph Theory

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• What a graph is ~9 min
• Vertices, edges, and order/size ~8 min
• Undirected vs. directed graphs ~9 min
• Simple graphs, multigraphs, and pseudographs ~10 min
• Weighted graphs ~8 min
• Degree, isolated, and pendant vertices, revisited ~9 min
• Subgraphs, induced subgraphs, and spanning subgraphs ~9 min
• Graph isomorphism, revisited ~10 min
• Graph representations: adjacency lists ~8 min
• Graph representations: adjacency and incidence matrices ~9 min

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• Walks, trails, and paths, revisited ~10 min
• Closed walks, circuits, and cycles, revisited ~10 min
• Connected components, revisited ~10 min
• Distance in a graph ~10 min
• Eccentricity ~8 min
• Diameter and radius ~9 min
• The center of a graph ~8 min
• Choosing the right connectivity vocabulary ~8 min

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• What a tree is, revisited ~9 min
• Equivalent characterizations of a tree ~12 min
• Leaves and internal vertices ~9 min
• Rooted trees and rooted-tree terminology ~10 min
• Ordered and binary trees, revisited ~10 min
• Spanning trees, structurally ~11 min
• The cut property, previewed ~9 min
• Counting spanning trees, revisited ~11 min
• Recognizing a tree by whichever property is easiest to check ~9 min

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• Breadth-first search ~10 min
• Depth-first search ~10 min
• BFS and shortest paths in unweighted graphs ~10 min
• DFS edge classification: tree edges and back edges ~9 min
• Traversal order and discovery/finish times ~10 min
• BFS vs. DFS, compared ~8 min
• Using traversal to test connectivity ~9 min
• Applications: maze-solving and dependency discovery ~10 min

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• Euler walks, trails, and circuits ~10 min
• The Königsberg bridges problem ~9 min
• Characterizing Eulerian graphs ~11 min
• Fleury's algorithm ~10 min
• Hierholzer's algorithm ~10 min
• Hamiltonian paths and cycles ~10 min
• Necessary conditions for Hamiltonicity ~10 min
• Dirac's and Ore's theorems ~11 min
• Eulerian vs. Hamiltonian, compared ~10 min
• The traveling salesperson problem, previewed ~9 min

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• Cut vertices ~10 min
• Bridges ~9 min
• Vertex cuts and edge cuts ~10 min
• Vertex connectivity and edge connectivity ~12 min
• k-connected graphs ~9 min
• Blocks ~10 min
• Menger's theorem ~12 min
• Network resilience as a connectivity question ~11 min

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• Proper vertex coloring, revisited ~10 min
• The chromatic number, revisited ~12 min
• Greedy coloring ~11 min
• Why greedy coloring isn't optimal ~10 min
• Brooks' theorem ~13 min
• Bipartite graphs and 2-colorability, revisited ~12 min
• Chromatic polynomials, revisited ~12 min
• Coloring as scheduling ~11 min

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• Planar graphs and plane embeddings ~10 min
• Faces of a plane graph ~9 min
• Euler's formula for planar graphs ~11 min
• Edge bounds for planar graphs ~10 min
• Kuratowski's theorem ~10 min
• K5 and K3,3 ~10 min
• Detecting nonplanarity ~9 min
• Dual graphs ~10 min
• The Platonic solids as planar graphs ~11 min
• The four-color theorem and the five-color theorem ~10 min

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• Weighted graphs, revisited ~14 min
• Dijkstra's algorithm ~18 min
• Why Dijkstra's algorithm fails on negative weights ~15 min
• Bellman-Ford's algorithm ~17 min
• The Floyd-Warshall algorithm ~17 min
• All-pairs shortest paths and distance matrices ~16 min
• Choosing a shortest-path algorithm ~14 min
• Shortest paths as a modeling tool ~16 min

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• The cut property, formalized ~12 min
• Kruskal's algorithm ~13 min
• Prim's algorithm ~13 min
• Borůvka's algorithm ~12 min
• Union-find and disjoint-set structures ~13 min
• Why the greedy choice works ~13 min
• Comparing MST algorithms ~10 min
• Network design as an MST problem ~12 min

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• Digraphs, revisited ~12 min
• Directed paths and directed cycles ~12 min
• Strong vs. weak connectivity ~12 min
• Strongly connected components ~15 min
• Condensation graphs ~12 min
• Directed acyclic graphs ~12 min
• Topological sorting ~16 min
• Dependency resolution and course prerequisites ~15 min
• Ranking a network: PageRank as a directed-graph problem ~16 min

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• Flow networks ~14 min
• Capacities, flow, and conservation ~16 min
• Residual graphs and augmenting paths ~16 min
• The Ford-Fulkerson algorithm ~18 min
• The max-flow min-cut theorem ~18 min
• Matching, revisited ~15 min
• Bipartite matching as a flow problem ~16 min
• Hall's theorem, revisited ~16 min
• General (non-bipartite) matching, previewed ~12 min
• Matching and flow in practice ~16 min

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• Comparing graph representations by cost ~9 min
• Algorithm complexity, revisited ~9 min
• O(V), O(E), and O(V+E) ~10 min
• P vs. NP, informally ~10 min
• NP-complete graph problems ~11 min
• The traveling salesperson problem as NP-hard ~11 min
• Why some graph problems resist every algorithm in this course ~10 min

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• Complete and complete bipartite graphs, revisited ~9 min
• Cycles, paths, and wheels ~9 min
• Regular graphs, revisited ~10 min
• The hypercube graph ~10 min
• The Petersen graph ~10 min
• Counting simple graphs ~10 min
• Degree sequences and graphical sequences, revisited ~11 min
• The complement of a graph ~9 min
• Graph operations: union, intersection, and line graphs ~10 min

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• The pigeonhole principle, revisited ~10 min
• Ramsey numbers ~12 min
• The party problem ~11 min
• R(3,3) ~13 min
• Monochromatic subgraphs in edge colorings ~12 min