Claude Berge

Bio: Claude Berge is an academic researcher from Princeton University. The author has contributed to research in topics: Hypergraph & Cubic graph. The author has an hindex of 22, co-authored 26 publications receiving 8938 citations.

The theory of graphs and its applications

Abstract: This book on the theory of graphs provides the reader with a mathematical tool which can be used in the behavioral sciences, in the theory of information, cybernetics, games, transport networks, as well as in set theory and matrix theory.

Hypergraphs: Combinatorics of Finite Sets

Abstract: 1. General Concepts. Dual Hypergraphs. Degrees. Intersecting Families. The Coloured Edge Property and Chvatal's Conjecture. The Helly Property. Section of a Hypergraph and the Kruskal-Katona Theorem. Conformal Hypergraphs. Representative Graphs. 2. Transversal Sets and Matchings. Transversal Hypergraphs. The Coefficients r and r'. r-Critical Hypergraphs. The Konig Property. 3. Fractional Transversals. Fractional Transversal Number. Fractional Matching of a Graph. Fractional Transversal Number of a Regularisable Hypergraph. Greedy Transversal Number. Ryser's Conjecture. Transversal Number of Product Hypergraphs. 4. Colourings. Chromatic Number. Particular Kinds of Colourings. Uniform Colourings. Extremal Problems Related to the Chromatic Number. Good Edge-Colourings of a Complete Hypergraph. An Application to an Extremal Problem. Kneser's Problem. 5. Hypergraphs Generalising Bipartite Graphs. Hypergraphs without Odd Cycles. Unimodular Hypergraphs. Balanced Hypergraphs. Arboreal Hypergraphs. Normal Hypergraphs. Mengerian Hypergraphs. Paranormal Hypergraphs. Appendix: Matchings and Colourings in Matroids. References.

Two theorems in graph theory

Abstract: Introduction. Given an unoriented graph (or 1-dimensional regular complex), let X be the set of all its vertices and U be the set of all its edges. When the graph is finite, the following problems arise: Problem 1: A set A c X is said to be internally stable if x e A, y e A implies (x, y) o U. The symbol A will denote the number of elements of A. Construct an internally stable set A such that A is maximum. Problem 2: A set B c X is said to be a cover if every edge of U is adjacent to at least one vertex in B. Construct a cover with the minimum number of elements. Problem 3: A set of edges V c U is said to be a matching if two edges of V have no vertex in common. Construct a matching with the maximum number of elements. A particular case of Problem 1 is the chess problem of Gauss: Put eight queens on the board such that no one can take any other. In n-person game theory, if the graph of domination is symmetrical, a maximum internally stable set turns out to be a maximum solution (in the von Neumann-Morgenstern sense1), and the more usual case can be solved by means of the Grundy functions.2 Problem 2 is the set theoretic dual of Problem 1, since the complement of an internally stable set is a cover, and conversely. Particular cases of Problem 3 are the problem of distinct representatives (P. Hall1) and the problem of Petersen (D. K6nig4). In the case where the graph is bipartite, Problem 3 has been solved by algebraic methods by 0. Ore,5 and an efficient algorithm has been given by H. Kuhn.6 Unfortunately, the linear programming duality used by H. Kuhn no longer subsists when the graph is not bipartite. (Note that Problem 2 is the linear program dual to Problem 3 in the bipartite case.) In view of solving the general case, this paper states two theorems: Theorem 1 gives a necessary and sufficient condition for recognizing whether a matching is maximum and provides an algorithm for Problem 3, while Theorem 2 yields an algorithm for Problems 1 and 2. The Theorems.-Consider a graph G = (X, U) with a matching V0; if u e Vo we shall say that edge u is strong, otherwise that u is weak. An alternating chain is a chain which does not use the same edge twice and is such that for any two adjacent edges one is strong and the other is weak. A vertex x which is not adjacent to a strong edge is said to be neutral, the set of all neutral points being N. We shall also consider a graph G constructed from G by adding a vertex a and connecting a to every neutral point with a strong edge. If there exists an alternating chain from a to a vertex x, we shall picture an arrow on the last edge (z, x), directed from z to x. A vertex x (a N) which is not adjacent to a directed edge is said to be inaccessible, the set of all inaccessible points being I. A vertex x (t N) adjacent to a weak edge directed to x and not to a strong edge directed to x is said to be weak, the set of all weak points being W. A vertex x (f N) adjacent to a strong edge directed to x and not to a weak edge directed to x is said to be strong,

A note on two problems in connexion with graphs

Abstract: We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to

The Design and Analysis of Computer Algorithms

Abstract: From the Publisher: With this text, you gain an understanding of the fundamental concepts of algorithms, the very heart of computer science. It introduces the basic data structures and programming techniques often used in efficient algorithms. Covers use of lists, push-down stacks, queues, trees, and graphs. Later chapters go into sorting, searching and graphing algorithms, the string-matching algorithms, and the Schonhage-Strassen integer-multiplication algorithm. Provides numerous graded exercises at the end of each chapter. 0201000296B04062001

Graph theory with applications

Abstract: (1977). Graph Theory with Applications. Journal of the Operational Research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.

Distributed algorithms

Abstract: In Distributed Algorithms, Nancy Lynch provides a blueprint for designing, implementing, and analyzing distributed algorithms. She directs her book at a wide audience, including students, programmers, system designers, and researchers. Distributed Algorithms contains the most significant algorithms and impossibility results in the area, all in a simple automata-theoretic setting. The algorithms are proved correct, and their complexity is analyzed according to precisely defined complexity measures. The problems covered include resource allocation, communication, consensus among distributed processes, data consistency, deadlock detection, leader election, global snapshots, and many others. The material is organized according to the system model-first by the timing model and then by the interprocess communication mechanism. The material on system models is isolated in separate chapters for easy reference. The presentation is completely rigorous, yet is intuitive enough for immediate comprehension. This book familiarizes readers with important problems, algorithms, and impossibility results in the area: readers can then recognize the problems when they arise in practice, apply the algorithms to solve them, and use the impossibility results to determine whether problems are unsolvable. The book also provides readers with the basic mathematical tools for designing new algorithms and proving new impossibility results. In addition, it teaches readers how to reason carefully about distributed algorithms-to model them formally, devise precise specifications for their required behavior, prove their correctness, and evaluate their performance with realistic measures. Table of Contents 1 Introduction 2 Modelling I; Synchronous Network Model 3 Leader Election in a Synchronous Ring 4 Algorithms in General Synchronous Networks 5 Distributed Consensus with Link Failures 6 Distributed Consensus with Process Failures 7 More Consensus Problems 8 Modelling II: Asynchronous System Model 9 Modelling III: Asynchronous Shared Memory Model 10 Mutual Exclusion 11 Resource Allocation 12 Consensus 13 Atomic Objects 14 Modelling IV: Asynchronous Network Model 15 Basic Asynchronous Network Algorithms 16 Synchronizers 17 Shared Memory versus Networks 18 Logical Time 19 Global Snapshots and Stable Properties 20 Network Resource Allocation 21 Asynchronous Networks with Process Failures 22 Data Link Protocols 23 Partially Synchronous System Models 24 Mutual Exclusion with Partial Synchrony 25 Consensus with Partial Synchrony