Abstract: : A knot in a directed graph is a useful concept in deadlock detection. This paper presents a distributed algorithm based on the work of Dijkstra and Scholten to identify knot in a graph by using a network of processes. (Author)

Abstract: This paper presents a formalization of the notion graphic. A graphic is considered to consist of an ordinary graph describing the overall structure and a set of attributes describing the shape, placement, etc. of the nodes and edges of the underlying graph. The formal handling of graphics is done by attributing the rules of graph grammars and by passing the attributes up and down the derivation tree of the graphic.

Abstract: The (d,k) graph problem which is a stiu open extremal problem in graph theory, has received very much attention from many authors due to its theoretic interest, and also due to its possible applications in communication network design. The problem consists in maximizing the number of nodes n of an undirected regular graph (d,k) of degree d and diameter k. In this paper, after a survey of the known results, we present two new families of graphs, and two methods of generating graphs given some existing ones, leading to further substantial improvements of some of the results gathered by Storwick [21] and recently improved by Arden and Lee [3] and also by Imase and Itoh [11].

Abstract: We introduce the concept of matching forests as a generalization of branchings in a directed graph and matchings in an undirected graph. Given special weights on the edges of a mixed graph, we present an efficient algorithm for finding an optimum weight-sum matching forest. The algorithm is a careful application of known branching and matching algorithms. The maximum cardinality matching forest problem is solved as a special case.

Abstract: This work presents a different approach to the continuing work done in the development of efficient algorithms for the solution of graph problems. While until now the main effort has been directed towards designing more efficient algorithms, we try to get a better solution by changing the graph representation. We suggest a new model for graph representation, which we named Graph Construction Representation. The new model proved to have two significant advantages over the conventional graph representations: (1) For many graphs, this model will use less space, or in other words will have a "succinct" representation. (2) We developed several algorithms which accept the new model as input. The time complexity for those algorithms when applied to graphs which are represented "succinctly" is improved in comparison with the best known algorithm. In this thesis the Graph Construction Representation is investigated. For many graph families it is shown how they are represented using this model. Bounds on the size of the representation are given. Several algorithms for checking graph properties are described, among them "Connectivity", "Has a Triangle?" and "Maximum Independent Set". We identify problems which are still open concerning the representation of graphs which are related to a graph already succinctly represented. We have only partial answers for these questions. We also present another graph representation, which was named the Small Circuit Representation, for which we prove that checking many graph properties is hard.

Abstract: The method of Lee and the simple application of graph theory seem to give different answers for flow networks. Lee's method is correct. The method whereby graph theory can be extended to give correct results is explained.

Abstract: This paper is a written version of the overview lecture on “Concurrency in Graph Grammars” given at the “2nd International Workshop on Graph Grammars and their Applications to Computer Science, 1982”. The intention of that lecture and this paper is to show that a number of results in the transformational theory of graph grammars can be considered already as contributions towards a theory of concurrency in graph grammars. Simulations of Petri nets within graph grammars are reviewed and simulations of algebraic specifications within graph grammars are introduced to cover also abstract data type concepts. More general concepts of concurrency are considered to be studied in the framework of graph grammars which go essentially beyond those in Petri nets. Finally it is proposed to combine the new approach of Ugo Montanari for synchronization with the aspects of concurrency reviewed above to obtain a graph grammar based model for concurrent and distributed systems.

Abstract: Programmed graph grammars are formally introduced and their generative power is investigated. Programmed graph grammars differ from other approaches to graph grammars in the so-called control diagram which controls the application order of productions. Restricting the form of the productions of a programmed graph grammar we get several classes of graph languages. These are compared mutually as well as with the hierarchy introduced by Nagl [18]. For unrestricted and monotone productions corresponding classes of graph languages coincide, while the class of context free programmed graph languages is properly contained in the class of context free graph languages in the sense of [18].

Abstract: This chapter reviews graph coloring problems at different levels of the complexity hierarchy and discusses their use to illustrate various complexity levels within the languages of graphs. The complexity terminology is fairly standard and follows closely that used by Garey and Johnson. The first problem of testing whether a graph is 2-colorable is introduced principally to clarify the notion space in nondeterministic machines. The second set of problems concern the number of colorings of a graph. The chapter discusses the creation of distinct recognition (= decision) problems associated with the well-known problem of finding the chromatic polynomial of a graph. Testing whether or not a graph is 2-colourable is one of the easiest algorithmic problems in the graph theory. The chromatic polynomial is also reviewed in the chapter.

Abstract: This survey (to be published in two parts in successive issues of the Bulletin) is based on lectures given at the St. Andrews Mathematical Colloquium in 1980. It attempts, as did the lectures, to indicate some of the achievements and problems of graph theory to mathematicians specializing in other subjects. It is therefore not aimed at experienced specialists in graph theory, although it might have some value for graduate students beginning work in the field. It makes no attempt to cover all important aspects of the subject or to achieve a fair balance between different ones, but emphasises, generally speaking, those topics which it was feasible to cover in the original lectures. One of my own reasons for becoming interested in graph theory was that I was intrigued by the possibility of developing non-trivial and fairly deep mathematics from a very simple initial concept. The ultimate illustration of this might be set theory, since sets are presumably the simplest and most basic of all mathematical structures and yet some aspects of set theory are both deep and substantial. If one wishes to move a little beyond set theory, which considers completely unstructured sets, one might start with sets carrying some structure, but only a very simply defined one. Arguably, the simplest structure which mathematicians commonly place on a set is a binary relation, whereby some pairs of elements are declared to be "related" in some sense. These will be unordered pairs if the binary relation is required to be symmetric—and, although this is by no means a necessary restriction, we will consider this case initially. A geometrical picture of the situation can be formed by regarding the elements of the underlying set as points (or "vertices") and considering two of these points to be joined by a line (or "edge") if they are related by our symmetric binary relation. The scope thus afforded for geometrical intuition is another of the attractions of graph theory. Our geometrical picture of a graph might suggest the following definitions. A graph is an ordered triple {V, E, I) such that

Abstract: The problem is considered of decomposing a given graph into the minimum number of complete subgraphs. Asymptotic results are obtained for the case where the graph is the complement of a graph with relatively few unisolated vertices.

Abstract: We give O(n) step algorithms for solving a number of graph problems on an n-&-times;n array of processors. The problems considered include: marking the bridges of an undirected graph, marking the articulation points of such a graph, finding the length of a shortest cycle, finding a minimum spanning tree, and a number of other problems.

Abstract: In this paper it is shown that R. Kowalski’s connection graph proof procedure /Ko75/ terminates with the empty clause for every unit-refutable clause set, provided an exhaustive search strategy is employed. This result holds for unrestricted tautology deletion, whereas subsumption requires certain precautions.

Abstract: We prove that there exists a one-to-one correspondence between the spanning trees and the fundamental cycle sets of a graph G if and only if G is 3-edge connected. Then we define a fundamental cycle set graph and prove that such a graph is a tree graph. It follows, therefore, that every fundamental cycle set graph on at least three vertices is Hamiltonian.

Abstract: Algorithms are considered to update graph properties after small changes have been made to a graph. Such update algorithms can use the approach either of directly determining the properties from the new graph or of actually updating the properties of the previous graph. Unfortunately, the execution times for the update algorithms developed by these two approaches often have the same worst-case order. One approach to comparing algorithms with equal worst-case orders is to count the frequency of execuuon for key operations, called active operations. For a class of algorithms that includes most update algorithms this technique can be extended to couniiiig the frequency of execution for active blocks of code. The approaches to the development of update algorithms and the use of active blocksin comparisons will be illustrated by the problem of updating ihe distance matrix of an undirected graph after an edge or vertex has been added or deleted.

Abstract: Let G[H] denote the composition of the graphs G and H. If G can be decomposed into one-factors and two-factors, H can be decomposed into one-factors, and H is not the empty graph on an odd number of vertices, then G[H] can be decomposed into one-factors.

Abstract: The notion of a finite automaton is perhaps the most basic notion of formal language theory. Each finite automaton is (can be represented by) a graph and it defines the set of strings referred to as the languageof the automaton. Thus we have here the situation where one graph is used to define a (possibly infinite) set of strings. In other words, one 'higher type" object is used to define a set of "lower type" objects certainly in general graphs have a more involved structure than strings. In this paper we present an attempt to develop the same type of methodology for defining graph languages. A hypergraph (see, e.g. [B] ), is a structure generalizing the notion of a graph. We will discuss the use of one hypergraph to define a (possibly infinite) set of graphs. (Actually the structure we use is more general than a hypergraph it is a hypergraph equipped with an extra graph structure.) Various grain-generating systems based on the use of hypergraphs are introduced and their properties (included the properties of classes of languages they define) are discussed. We point out also the relationship of these systems to the theory of Petri nets. The paper presents only part of the results concerning the theory of hypergraph-based systems.

Abstract: Directed graphs have been used as the theoretical base for many applications in Computer Science -- especially artificial intelligence. Two drawbacks to the use of directed graphs for more practical work have been:1. The difficulty in associating data of a variety of types with the nodes in the graph and with the connections between the nodes, and2. The inability to search and modify the graph efficiently as the size and complexity of the graph grows.This paper describes a library of cooperating routines for dealing with directed graphs that overcomes these objections. The first problem is overcome by adding to the usual directed graph model the concepts of "data tags" and "connection descriptors." The problem of efficiency is addressed by using a binary tree facility to implement the augmented directed graph. This paper describes the augmented directed graph "access method" and how it can be used as the base for application development.