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Издательство Springer, 2007, -772 pp.Graph theory is a very popular area of discrete mathematics with not only numerous theoretical developments, but also countless applications to practical problems. As a research area, graph theory is still relatively young, but it is maturing rapidly with many deep results having been discovered over the last couple of decades.

The theory of graphs can be roughly partitioned into two branches: the areas of undirected graphs and directed graphs (digraphs). Even though both areas have numerous important applications, for various reasons, undirected graphs have been studied much more extensively than directed graphs. One of the reasons is that undirected graphs form in a sense a special class of directed graphs (symmetric digraphs) and hence problems that can be formulated for both directed and undirected graphs are often easier for the latter. Another reason is that, unlike for the case of undirected graphs, for which there are several important books covering both classical and recent results, no previous book covers more than a small fraction of the results obtained on digraphs within the last 25 years. Typically, digraphs are considered only in one chapter or by a few elementary results scattered throughout the book.

Despite all this, the theory of directed graphs has developed enormously within the last three decades. There is an extensive literature on digraphs (more than 3000 papers). Many of these papers contain, not only interesting theoretical results, but also important algorithms as well as applications. This clearly indicates a real necessity for a book, covering not only the basics on digraphs, but also deeper, theoretical as well as algorithmic, results and applications.

The present book is an attempt to ¯ll this huge gap in the literature and may be considered as a handbook on the subject. It starts at a level that can be understood by readers with only a basic knowledge in university mathematics and goes all the way up to the latest research results in several areas (including connectivity, orientations of graphs, submodular flows, paths and cycles in digraphs, generalizations of tournaments and generalizations of digraphs). The book contains more than 700 exercises and a number of applications as well as sections on highly applicable subjects. Due to the fact that we wish to address different groups of readers (advanced undergraduate and graduate students, researchers in discrete mathematics and researchers in various areas including computer science, operations research, artificial intelligence, social sciences and engineering) not all topics will be equally interesting to all potential readers. However, we strongly believe that all readers will ¯nd a number of topics of special interest to them.

Even though this book should not be seen as an encyclopedia on directed graphs, we included as many interesting results as possible. The book contains a considerable number of proofs, illustrating various approaches and techniques used in digraph theory and algorithms.

One of the main features of this book is the strong emphasis on algorithms. This is something which is regrettably omitted in some books on graphs. Algorithms on (directed) graphs often play an important role in problems arising in several areas, including computer science and operations research. Secondly, many problems on (directed) graphs are inherently algorithmic. Hence, whenever possible we give constructive proofs of the results in the book. From these proofs one can very often extract an efficient algorithm for the problem studied. Even though we describe many algorithms, partly due to space limitations, we do not supply all the details necessary in order to implement these algorithms. The later (often highly non-trivial step) is a science in itself and we refer the reader to books on data structures.

Another important feature is the vast number of exercises which not only help the reader to train his or her understanding of the material, but also complements the results introduced in the text by covering even more material. Attempting these exercises (most of which appear in a book for the first time) will help the reader to master the subject and its main techniques. Through its broad coverage and the exercises, stretching from easy to quite difficult, the book will be useful for courses on subjects such as (di)graph theory, combinatorial optimization and graph algorithms. Furthermore, it can be used for more focused courses on topics such as flows, cycles and connectivity. The book contains a large number of illustrations. This will help the reader to understand otherwise difficult concepts and proofs. To facilitate the use of this book as a reference book and as a graduate textbook, we have added comprehensive symbol and subject indexes. It is our hope that the detailed subject index will help many readers to find what they are looking for without having to read through whole chapters. In particular, there are entries for open problems and conjectures. Every class of digraphs which is studied in the book has its own entry containing the majority of pages on which this class is treated. As sub-entries to the entry `proof techniques' we have indexed different proof techniques and some representative pages where the technique is illustrated. Due to our experience, we think that the book will be a useful teaching and reference resource for several decades to come.

In this book we cover the majority of the important topics on digraphs ranging from quite elementary to very advanced ones. Below we give a brief outline of some of the main highlights of this book. Readers who are looking for more detailed information are advised to consult the list of contents or the subject index at the end of the book.Basic Terminology, Notation and Results

Distances

Flows in Networks

Classes of Digraphs

Hamiltonicity and Related Problems

Hamiltonian Refinements

Global Connectivity

Orientations of Graphs

Disjoint Paths and Trees

Cycle Structure of Digraphs

Generalizations of Digraphs

Additional Topics

The theory of graphs can be roughly partitioned into two branches: the areas of undirected graphs and directed graphs (digraphs). Even though both areas have numerous important applications, for various reasons, undirected graphs have been studied much more extensively than directed graphs. One of the reasons is that undirected graphs form in a sense a special class of directed graphs (symmetric digraphs) and hence problems that can be formulated for both directed and undirected graphs are often easier for the latter. Another reason is that, unlike for the case of undirected graphs, for which there are several important books covering both classical and recent results, no previous book covers more than a small fraction of the results obtained on digraphs within the last 25 years. Typically, digraphs are considered only in one chapter or by a few elementary results scattered throughout the book.

Despite all this, the theory of directed graphs has developed enormously within the last three decades. There is an extensive literature on digraphs (more than 3000 papers). Many of these papers contain, not only interesting theoretical results, but also important algorithms as well as applications. This clearly indicates a real necessity for a book, covering not only the basics on digraphs, but also deeper, theoretical as well as algorithmic, results and applications.

The present book is an attempt to ¯ll this huge gap in the literature and may be considered as a handbook on the subject. It starts at a level that can be understood by readers with only a basic knowledge in university mathematics and goes all the way up to the latest research results in several areas (including connectivity, orientations of graphs, submodular flows, paths and cycles in digraphs, generalizations of tournaments and generalizations of digraphs). The book contains more than 700 exercises and a number of applications as well as sections on highly applicable subjects. Due to the fact that we wish to address different groups of readers (advanced undergraduate and graduate students, researchers in discrete mathematics and researchers in various areas including computer science, operations research, artificial intelligence, social sciences and engineering) not all topics will be equally interesting to all potential readers. However, we strongly believe that all readers will ¯nd a number of topics of special interest to them.

Even though this book should not be seen as an encyclopedia on directed graphs, we included as many interesting results as possible. The book contains a considerable number of proofs, illustrating various approaches and techniques used in digraph theory and algorithms.

One of the main features of this book is the strong emphasis on algorithms. This is something which is regrettably omitted in some books on graphs. Algorithms on (directed) graphs often play an important role in problems arising in several areas, including computer science and operations research. Secondly, many problems on (directed) graphs are inherently algorithmic. Hence, whenever possible we give constructive proofs of the results in the book. From these proofs one can very often extract an efficient algorithm for the problem studied. Even though we describe many algorithms, partly due to space limitations, we do not supply all the details necessary in order to implement these algorithms. The later (often highly non-trivial step) is a science in itself and we refer the reader to books on data structures.

Another important feature is the vast number of exercises which not only help the reader to train his or her understanding of the material, but also complements the results introduced in the text by covering even more material. Attempting these exercises (most of which appear in a book for the first time) will help the reader to master the subject and its main techniques. Through its broad coverage and the exercises, stretching from easy to quite difficult, the book will be useful for courses on subjects such as (di)graph theory, combinatorial optimization and graph algorithms. Furthermore, it can be used for more focused courses on topics such as flows, cycles and connectivity. The book contains a large number of illustrations. This will help the reader to understand otherwise difficult concepts and proofs. To facilitate the use of this book as a reference book and as a graduate textbook, we have added comprehensive symbol and subject indexes. It is our hope that the detailed subject index will help many readers to find what they are looking for without having to read through whole chapters. In particular, there are entries for open problems and conjectures. Every class of digraphs which is studied in the book has its own entry containing the majority of pages on which this class is treated. As sub-entries to the entry `proof techniques' we have indexed different proof techniques and some representative pages where the technique is illustrated. Due to our experience, we think that the book will be a useful teaching and reference resource for several decades to come.

In this book we cover the majority of the important topics on digraphs ranging from quite elementary to very advanced ones. Below we give a brief outline of some of the main highlights of this book. Readers who are looking for more detailed information are advised to consult the list of contents or the subject index at the end of the book.Basic Terminology, Notation and Results

Distances

Flows in Networks

Classes of Digraphs

Hamiltonicity and Related Problems

Hamiltonian Refinements

Global Connectivity

Orientations of Graphs

Disjoint Paths and Trees

Cycle Structure of Digraphs

Generalizations of Digraphs

Additional Topics

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