Jörgen Bang-Jensen and Gregory Gutin. Digraphs. Theory, Algorithms and Applications. Springer

Transcription

1 Jörgen Bang-Jensen and Gregory Gutin Digraphs Theory, Algorithms and Applications Springer

2 Contents 1. Basic Terminology, Notation and Results Sets, Subsets, Matrices and Vectors Digraphs, Subdigraphs, Neighbours, Degrees Isomorphism and Basic Operations on Digraphs Walks, Trails, Paths, Cycles and Path-Cycle Subdigraphs Strong and Unilateral Connectivity Undirected Graphs, Biorientations and Orientations Mixed Graphs and Hypergraphs Classes of Directed and Undirected Graphs Algorithmic Aspects Algorithms and their Complexity A/^-Complete and M"P-Eaxd Problems Application: Solving the 2-Satisfiability Problem Exercises Distances Terminology and Notation on Distances Structure of Shortest Paths Algorithms for Finding Distances in Digraphs Breadth-First Search (BFS) Acyclic Digraphs Dijkstra's Algorithm The Bellman-Ford-Moore Algorithm The Floyd-Warshall Algorithm Inequalities Between Radius, Out-Radius and Diameter Radius and Diameter of a Strong Digraph Extreme Values of Out-Radius and Diameter Maximum Finite Diameter of Orientations Minimum Diameter of Orientations of Multigraphs Minimum Diameter Orientations of Complete Multipartite Graphs Minimum Diameter Orientations of Extensions of Graphs Minimum Diameter Orientations of Cartesian Products of Graphs 71

3 xvi Contents 2.10 Kings in Digraphs Kings in Tournaments Kings in Semicomplete Multipartite Digraphs Kings in Generalizations of Tournaments Application: The One-Way Street and the Gossip Problems The One-Way Street Problem and Orientations of Digraphs The Gossip Problem Application: Exponential Neighbourhood Local Search for the TSP Local Search for the TSP Linear Time Searchable Exponential Neighbourhoods for the TSP The Assignment Neighbourhoods Diameters of Neighbourhood Structure Digraphs for the TSP Exercises Flows in Networks Definitions and Basic Properties Flows and Their Balance Vectors The Residual Network Reductions Among Different Flow Models Eliminating Lower Bounds Flows with one Source and one Sink Circulations Networks with Bounds and Costs on the Vertices Flow Decompositions Working with the Residual Network The Maximum Flow Problem The Ford-Fulkerson Algorithm Maximum Flows and Linear Programming Polynomial Algorithms for Finding a Maximum (s,t)-flow Flow Augmentations Along Shortest Augmenting Pathsll Blocking Flows in Layered Networks and Dinic's Algorithm The Preflow-Push Algorithm Unit Capacity Networks and Simple Networks Unit Capacity Networks Simple Networks Circulations and Feasible Flows Minimum Value Feasible (s, t)-flows Minimum Cost Flows Characterizing Minimum Cost Flows Building up an Optimal Solution 134

4 Contents xvii 3.11 Applications of Flows Maximum Matchings in Bipartite Graphs The Directed Chinese Postman Problem Finding Subdigraphs with Prescribed Degrees Path-Cycle Factors in Directed Multigraphs Cycle Subdigraphs Covering Specified Vertices The Assignment Problem and the Transportation Problem Exercises Classes of Digraphs Depth-First Search Acyclic Orderings of the Vertices in Acyclic Digraphs Transitive Digraphs, Transitive Closures and Reductions Strong Digraphs Line Digraphs The de Bruijn and Kautz Digraphs and their Generalizations Series-Parallel Digraphs Quasi-Transitive Digraphs The Path-Merging Property and Path-Mergeable Digraphs Locally In-Semicomplete and Locally Out-Semicomplete Digraphs Locally Semicomplete Digraphs Round Digraphs Non-Strong Locally Semicomplete Digraphs Strong Round Decomposable Locally Semicomplete Digraphs Classification of Locally Semicomplete Digraphs Totally ^-Decomposable Digraphs Intersection Digraphs Planar Digraphs Application: Gaussian Elimination Exercises Hamiltonicity and Related Problems Necessary Conditions for Hamiltonicity of Digraphs Path-Contraction Quasi-Hamiltonicity Pseudo-Hamiltonicity and 1-Quasi-Hamiltonicity Algorithms for Pseudo- and Quasi-Hamiltonicity Path Covering Number Path Factors of Acyclic Digraphs with Applications Hamilton Paths and Cycles in Path-Mergeable Digraphs Hamilton Paths and Cycles in Locally In-Semicomplete Digraphs Hamilton Cycles and Paths in Degree-Constrained Digraphs. 240

5 xviii Contents Sufficient Conditions The Multi-Insertion Technique Proofs of Theorems and Longest Paths and Cycles in Semicomplete Multipartite Digraphs Basic Results The Good Cycle Factor Theorem Consequences of Lemma Yeo's Irreducible Cycle Subdigraph Theorem and its Applications Longest Paths and Cycles in Extended Locally Semicomplete Digraphs Hamilton Paths and Cycles in Quasi-Transitive Digraphs Vertex-Heaviest Paths and Cycles in Quasi-Transitive Digraphs Hamilton Paths and Cycles in Various Classes of Digraphs Exercises Hamiltonian Refinements Hamiltonian Paths with a Prescribed End-Vertex Weakly Hamiltonian-Connected Digraphs Results for Extended Tournaments Results for Locally Semicomplete Digraphs Hamiltonian-Connected Digraphs Finding a Hamiltonian (x, j/)-path in a Semicomplete Digraph Pancyclicity of Digraphs (Vertex-)Pancyclicity in Degree-Constrained Digraphs Pancyclicity in Extended Semicomplete and Quasi- Transitive Digraphs Pancyclic and Vertex-Pancyclic Locally Semicomplete Digraphs Further Pancyclicity Results Cycle Extendability in Digraphs Arc-Pancyclicity Hamiltonian Cycles Containing or Avoiding Prescribed Arcs Hamiltonian Cycles Containing Prescribed Arcs Avoiding Prescribed Arcs with a Hamiltonian Cycle Hamiltonian Cycles Avoiding Arcs in 2-Cycles Arc-Disjoint Hamiltonian Paths and Cycles Oriented Hamiltonian Paths and Cycles Covering All Vertices of a Digraph by Few Cycles Cycle Factors with a Fixed Number of Cycles The Effect of a(d) on Spanning Configurations of Paths and Cycles Minimum Strong Spanning Subdigraphs A Lower Bound for General Digraphs 331

6 Contents xix The MSSS Problem for Extended Semicomplete Digraphs The MSSS Problem for Quasi-Transitive Digraphs The MSSS Problem for Decomposable Digraphs Application: Domination Number of TSP Heuristics Exercises Global Connectivity Additional Notation and Preliminaries The Network Representation of a Directed Multigraph Ear Decompositions Menger's Theorem Application: Determining Arc- and Vertex-Strong Connectivity The Splitting off Operation Increasing the Arc-Strong Connectivity Optimally Increasing the Vertex-Strong Connectivity Optimally One-Way Pairs Optimal fc-strong Augmentation Special Classes of Digraphs Splittings Preserving fc-strong Connectivity A Generalization of Arc-Strong Connectivity Arc Reversals and Vertex-Strong Connectivity Minimally fc-(arc)-strong Directed Multigraphs Minimallyfc-Arc-StrongDirected Multigraphs Minimally fc-strong Digraphs Critically fc-strong Digraphs Arc-Strong Connectivity and Minimum Degree Connectivity Properties of Special Classes of Digraphs Highly Connected Orientations of Digraphs Packing Cuts Application: Small Certificates for fc-(arc)-strong Connectivity Finding Small Certificates for Strong Connectivity Finding fc-strong Certificates for fc > Certificates forfc-arc-strongconnectivity Exercises Orientations of Graphs Underlying Graphs of Various Classes of Digraphs Underlying Graphs of Transitive and Quasi-Transitive Digraphs Underlying Graphs of Locally Semicomplete Digraphs Local Tournament Orientations of Proper Circular Arc Graphs Underlying Graphs of Locally In-Semicomplete Digraphs Fast Recognition of Locally Semicomplete Digraphs 429

7 XX Contents 8.3 Orientations With no Even Cycles Colourings and Orientations of Graphs Orientations and Nowhere Zero Integer Flows Orientations Achieving High Arc-Strong Connectivity Orientations Respecting Degree Constraints Orientations with Prescribed Degree Sequences Restrictions on Subsets of Vertices Submodular Flows Submodular Flow Models Existence of Feasible Submodular Flows Minimum Cost Submodular Flows Applications of Submodular Flows Orientations of Mixed Graphs Exercises 467 Disjoint Paths and Trees Additional Definitions Disjoint Path Problems The Complexity of the Jfc-Path Problem Sufficient Conditions for a Digraph to befc-linked The fc-path Problem for Acyclic Digraphs Linkings in Tournaments and Generalizations of Tournaments Sufficient Conditions in Terms of (Local-)Connectivity The 2-Path Problem for Semicomplete Digraphs The 2-Path Problem for Generalizations of Tournaments Linkings in Planar Digraphs Arc-Disjoint Branchings Implications of Edmonds' Branching Theorem Edge-Disjoint Mixed Branchings Arc-Disjoint Path Problems Arc-Disjoint Paths in Acyclic Directed Multigraphs Arc-Disjoint Paths in Eulerian Directed Multigraphs Arc-Disjoint Paths in Tournaments and Generalizations of Tournaments Integer Multicommodity Flows Arc-Disjoint In- and Out-Branchings Minimum Cost Branchings Matroid Intersection Formulation An Algorithm for a Generalization of the Min Cost Branching Problem The Minimum Covering Arborescence Problem Increasing Rooted Arc-Strong Connectivity by Adding New Arcs Exercises 538

8 Contents XXI 10. Cycle Structure of Digraphs Vector Spaces of Digraphs Polynomial Algorithms for Paths and Cycles Disjoint Cycles and Feedback Sets Complexity of the Disjoint Cycle and Feedback Set Problems Disjoint Cycles in Digraphs with Minimum Out-Degree at Least A; Feedback Sets and Linear Orderings in Digraphs Disjoint Cycles Versus Feedback Sets Relations Between Parameters щ and T* Solution of Younger's Conjecture Application: The Period of Markov Chains Cycles of Length к Modulo p Complexity of the Existence of Cycles of Length к Modulo p Problems Sufficient Conditions for the Existence of Cycles of Length к Modulo p 'Short' Cycles in Semicomplete Multipartite Digraphs Cycles Versus Paths in Semicomplete Multipartite Digraphs Girth Additional Topics on Cycles Chords of Cycles Adam's Conjecture 584 lo.hexercises Generalizations of Digraphs Properly Coloured Trails in Edge-Coloured Multigraphs Properly Coloured Euler Trails Properly Coloured Cycles Connectivity of Edge-Coloured Multigraphs Alternating Cycles in 2-Edge-Coloured Bipartite Multigraphs Longest Alternating Paths and Cycles in 2-Edge-Coloured Complete Multigraphs Properly Coloured Hamiltonian Paths in c-edge-coloured Complete Graphs, с > Properly Coloured Hamiltonian Cycles in c-edge-coloured Complete Graphs, с > Arc-Coloured Directed Multigraphs Complexity of the Alternating Directed Cycle Problem The Functions /(n) and g(n) Weakly Eulerian Arc-Coloured Directed Multigraphs Hypertournaments Out-Degree Sequences of Hypertournaments 628

9 xxii Contents Hamilton Paths Hamilton Cycles Application: Alternating Hamilton Cycles in Genetics Proof of Theorem Proof of Theorem Exercises Additional Topics Seymour's Second Neighbourhood Conjecture Ordering the Vertices of a Digraph of Paired Comparisons Paired Comparison Digraphs The Капо-Sakamoto Methods of Ordering Orderings for Semicomplete PCDs The Mutual Orderings Complexity and Algorithms for Forward and Backward Orderings (fc,z)-kernels Kernels Quasi-Kernels List Edge-Colourings of Complete Bipartite Graphs Homomorphisms - A Generalization of Colourings Other Measures of Independence in Digraphs Matroids The Dual of a Matroid The Greedy Algorithm for Matroids Independence Oracles Union of Matroids Two Matroid Intersection Intersections of Three or More Matroids Finding Good Solutions to A/7>-Hard Problems Exercises 677 References 683 Symbol Index 717 Author Index 723 Subject Index 731