Digraphs: Theory, Algorithms and Applications

ISBN-13: 9781848009974 / Angielski / Twarda / 2009 / 795 str.

Jorgen Bang-Jensen; Gregory Gutin

Opis

The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

Recenzje

From the reviews:

"Digraphs (directed graphs) are a long-standing and important field of graph theory. Nevertheless, this is the first comprehensive monograph devoted to the subject. As a handbook, it addresses various groups of readers (students as well as researchers, and from various areas of applications). Throughout, because of their importance for applications, emphasis is on algorithms (in form of constructive proofs whenever possible). ... a valuable and indispensable reference for many years to come. Readers and users will be thankful!" (P. Schmitt, Monatshefte für Mathematik, Vol. 141 (1), 2004)

From the reviews of the second edition:

"This is a very comprehensive volume on directed graphs and related topics. ... an excellent reference. ... the book will be useful not just for researchers focusing on digraphs, but also for everyone ... . an extensive list of conjectures and open questions is included in every chapter. These are typically very well presented and easy to understand." (Miklós Bóna, MAA Online, February, 2009)

"The first edition of this book was published in 2000 ... . The area has continued to develop and this reorganized and updated edition contains an additional six chapters ... . there is a strong emphasis on algorithms, and constructive proofs are given whenever possible. There are more than 650 exercises and 150 open problems and conjectures. ... the bibliography contains 934 items and the author index contains 740 names. This book would be a valuable resource for anyone interested in digraphs and their applications." (J. W. Moon, Zentralblatt MATH, Vol. 1170, 2009)

Spis treści

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.

Blurb

The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity.

Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems.

As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem.

Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject.

Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

Jørgen Bang-Jensen is a Professor in the Department of Mathematics and Computer Science at the University of Southern Denmark, Odense, Denmark.

Gregory Gutin is Professor of Computer Science at Royal Holloway College, University of London, UK.