Defines and explores the implementation and figures of the algorithms required for various applications, offering commentary, descriptions, and exercises for developers, researchers, and students. Graph algorithms are critical for a range of applications, such as network connectivity, circuit design, scheduling, transaction processing, and resource allocation. This work presents many algorithms and their explanations. It also includes detailed figures, with accompanying commentary. Graph Algorithms Graph Properties and Types 3(72) Glossary 7(9) Graph ADT 16(5) Adjacency-Matrix Representation 21(6) Adjacency-Lists Representation 27(4) Variations, Extensions, and Costs 31(9) Graph Generators 40(10) Simple, Euler, and Hamilton Paths 50(14) Graph-Processing Problems 64(11) Graph Search 75(66) Exploring a Maze 76(5) Depth-First Search 81(5) Graph-Search ADT Functions 86(5) Properties of DFS Forests 91(8) DFS Algorithms 99(7) Separability and Biconnectivity 106(8) Breadth-First Search 114(10) Generalized Graph Search 124(9) Analysis of Graph Algorithms 133(8) Digraphs and DAGs 141(78) Glossary and Rules of the Game 144(8) Anatomy of DFS in Digraphs 152(9) Reachability and Transitive Closure 161(13) Equivalence Relations and Partial Orders 174(4) DAGs 178(5) Topological Sorting 183(10) Reachability in DAGs 193(3) Strong Components in Digraphs 196(12) Transitive Closure Revisited 208(4) Perspective 212(7) Minimum Spanning Trees 219(46) Representations 222(6) Underlying Principles of MST Algorithms 228(7) Prim's Algorithm and Priority-First Search 235(11) Kruskal's Algorithm 246(6) Boruvka's Algorithm 252(3) Comparisons and Improvements 255(6) Euclidean MST 261(4) Shortest Paths 265(88) Underlying Principles 273(7) Dijkstra's algorithm 280(10) All-Pairs Shortest Paths 290(10) Shortest Paths in Acyclic Networks 300(8) Euclidean Networks 308(6) Reduction 314(17) Negative Weights 331(19) Perspective 350(3) Network Flows 353(120) Flow Networks 359(11) Augmenting-Path Maxflow Algorithms 370(26) Preflow-Push Maxflow Algorithms 396(15) Maxflow Reductions 411(18) Mincost Flows 429(10) Network Simplex Algorithm 439(18) Mincost-Flow Reductions 457(10) Perspective 467(6) References for Part Five 473(2) Index 475