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.

Digraphs Theory Algorithms And Applications

Publisher Summary
This chapter focuses on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained. As a formalism, rewrite systems have the full power of Turing machines and may be thought of as nondeterministic Markov algorithms over terms rather than strings. The theory of rewriting is in essence a theory of normal forms. To some extent, it is an outgrowth of the study of A. Church's Lambda Calculus and H. B. Curry's Combinatory Logic. The chapter discusses the syntax and semantics of equations from the algebraic, logical, and operational points of view. To use a rewrite system as a decision procedure, it must be convergent. The chapter describes this fundamental concept as an abstract property of binary relations. To use a rewrite system for computation or as a decision procedure for validity of identities, the termination property is crucial. The chapter presents the basic methods for proving termination. The chapter discusses the question of satisfiability of equations and the convergence property applied to rewriting.

CHAPTER 6 – Rewrite Systems

This is a reissue of the book Tree Automata by F G\'ecseg and M Steinby originally published in 1984 by Akad\'emiai Kiad\'o, Budapest Some mistakes have been corrected and a few obscure passages have been clarified Moreover, some more recent contributions and current lines of research are reviewed in an appendix that also contains several new references

Tree automata

This book provides the most basic problems, concepts, and well-established results from the topological structure and analysis of interconnection networks in the graph-theoretic language. It covers the basic principles and methods of network design, several well-known networks such as hypercubes, de Bruijn digraphs, Kautz digraphs, double loop, and other networks, and the newest parameters to measure performance of fault-tolerant networks such as Menger number, Rabin number, fault-tolerant diameter, wide-diameter, restricted connectivity, and (l,w)-dominating number. Audience: The book is suitable for those readers who are working on or intend to start research in design analysis of the topological structure of interconnection networks, particularly undergraduates and postgraduates specializing in computer science and applied mathematics.

Topological Structure and Analysis of Interconnection Networks

Fundamental properties of infinite trees

There is diverse literature on various properties of a class of graphs known as circulants. We present a new result which answers the previously unsolved question of characterizing the connection sequence of circulants having point connectivity equal to point degree. We also develop some theorems regarding a new generalization of connectivity known as super-connectivity. In addition, we give a survey of published results pertinent to the study of connectivity of circulants.

Circulants and their connectivities

On the algebraic structure of rooted trees

http://www.jstor.org/stable/pdfplus/2321858.pdf

Robbins's theorem for mixed multigraphs

It is shown that a connected graph G spans an eulerian graph if and only if G is not spanned by an odd complete bigraph K(2m + 1, 2n + 1). A disconnected graph spans an eulerian graph if and only if it is not the union of the trivial graph with a complete graph of odd order. Exact formulas are obtained for the number of lines which must be added to such graphs in order to get eulerian graphs.

The spanning subgraphs of eulerian graphs

This paper presents some results regarding the design of reliable networks. The problem under consideration involves networks which are undirected graphs having equal and independent edge failure probabilities. The index of reliability is the probability that the network fails (becomes disconnected). For “small” edge failure probabilities and given p and q there exists a class of p vertex, q edge graphs with the property that any graph in the class has a smaller probability of disconnection than any graph outside of the class. We solve the problem of synthesizing graphs in this class.

Ralph Tindell

Papers

Circulants and their connectivities

On the algebraic structure of rooted trees

Robbins's theorem for mixed multigraphs

The spanning subgraphs of eulerian graphs

Combinatorial optimization problems in the analysis and design of probabilistic networks