内容

目次

1. Basic Terminology, Notation and Results. -1.1 Sets, Matrices and Vectors. -1.2 Digraphs, Subdigraphs, Neighbours, Degrees. -1.3 Isomorphism and Basic Operations on Digraphs. -1.4 Walks, Trails, Paths, Cycles and Path-Cycle Subdigraphs. -1.5 Strong and Unilateral Connectivity. -1.6 Undirected Graphs, Biorientations and Orientations. -1.7 Trees and Euler Trails in Digraphs. -1.8 Mixed Graphs, Orientations of Digraphs, and Hypergraphs. -1.9 Depth-First Search. -1.10 Exercises. -2. Classes of Digraphs. -2.1 Acyclic Digraphs. -2.2 Multipartite Digraphs and Extended Digraphs. -2.3 Transitive Digraphs, Transitive Closures and Reductions. -2.4 Line Digraphs. -2.5 The de Bruijn and Kautz Digraphs. -2.6 Series-Parallel Digraphs. -2.7 Quasi-Transitive Digraphs. -2.8 Path-Mergeable Digraphs. -2.9 Locally In/Out-Semicomplete Digraphs. -2.10 Locally Semicomplete Digraphs. -2.11 Totally F-Decomposable Digraphs. -2.12 Planar Digraphs. -2.13 Digraphs of Bounded Tree-Width -2.14 Other Families of Digraphs. -2.15 Exercises. -3. Distances. -3.1 Terminology and Notation on Distances. -3.2 Structure of Shortest Paths. -3.3 Algorithms for Finding Distances in Digraphs. -3.3.1 Breadth-First Search (BFS). -3.3.2 Acyclic Digraphs. -3.3.3 Dijkstra’s Algorithm. -3.3.4 The Bellman-Ford-Moore Algorithm. -3.3.5 The Floyd-Warshall Algorithm. -3.4 Inequalities on Diameter. -3.5 Minimum Diameter of Orientations of Multigraphs. -3.6 Minimum Diameter Orientations of Some Graphs and Digraphs. -3.7 Kings in Digraphs. -3.8 (k, l)-Kernels. -3.9 Exercises. -4. Flows in Networks. -4.1 Definitions and Basic Properties. -4.2 Reductions Among Different Flow Models. -4.3 Flow Decompositions. -4.4 Working with the Residual Network. -4.5 The Maximum Flow Problem. -4.6 Polynomial Algorithms for Finding a Maximum (s, t)-Flow. -4.7 Unit Capacity Networks and Simple Networks. -4.8 Circulations and Feasible Flows. -4.9 Minimum Value Feasible (s, t)-Flows. -4.10 Minimum Cost Flows. -4.11 Applications of Flows.-4.12 Exercises. -5. Connectivity of Digraphs. -5.1 Additional Notation and Preliminaries. -5.2 Finding the Strong Components of a Digraph. -5.3 Ear Decompositions. -5.4 Menger’s Theorem. -5.5 Determining Arc- and Vertex-Strong Connectivity. -5.6 Minimally k-(Arc)-Strong Directed Multigraphs. -5.7 Critically k-Strong Digraphs. -5.8 Connectivity Properties of Special Classes of Digraphs. -5.9 Disjoint X-paths in Digraphs. -5.10 Exercises. -6. Hamiltonian, Longest and Vertex-Cheapest Paths and Cycles. -6.1 Complexity. -6.2 Hamilton Paths and Cycles in Path-Mergeable Digraphs. -6.3 Hamilton Paths and Cycles in Locally In-Semicomplete Digraphs. -6.4 Hamilton Cycles and Paths in Degree-Constrained Digraphs. -6.5 Longest Paths and Cycles in Degree-Constrained Oriented Graphs. -6.6 Longest Paths and Cycles in Semicomplete Multipartite Digraphs. -6.7 Hamilton Paths and Cycles in Quasi-Transitive Digraphs. -6.8 Vertex-Cheapest Paths and Cycles. -6.9 Hamilton Paths and Cycles in Various Classes of Digraphs. -6.10 Exercises. -7. Restricted Hamiltonian Paths and Cycles. -7.1 Hamiltonian Paths with a Prescribed End-Vertex. -7.2 Weakly Hamiltonian-Connected Digraphs. -7.3 Hamiltonian-Connected Digraphs. -7.4 Hamiltonian Cycles Containing or Avoiding Prescribed Arcs. -7.5 Arc-Traceable Digraphs. -7.6 Oriented Hamiltonian Paths and Cycles. -7.7 Exercises. -8. Paths and Cycles of Prescribed Lengths. -8.1 Pancyclicity of Digraphs. -8.2 Colour Coding: Efficient Algorithms for Paths and Cycles. -8.3 Cycles of Length k Modulo p. -8.4 Girth. -8.5 Short Cycles in Semicomplete Multipartite Digraphs. -8.6 Exercises. -9. Branchings. -9.1 Tutte’s Matrix Tree Theorem. -9.2 Optimum Branchings. -9.3 Arc-Disjoint Branchings. -9.4 Implications of Edmonds’ Branching Theorem. -9.5 Out-Branchings with Degree Bounds. -9.6 Arc-Disjoint In- and Out-Branchings. -9.7 Out-Branchings with Extremal Number of Leaves. -9.8 The Source Location Problem. -9.9 Miscellaneous Topics. -9.10 Exercises. -10.