Showing papers on "Connectivity published in 1987"
TL;DR: This paper investigates several problems concerning the existence and enumeration of highly irregular graphs as well as their independence numbers, with particular focus on the corresponding problems for highly irregular trees.
Abstract: A connected graph is highly irregular if each of its vertices is adjacent only to vertices with distinct degrees. In this paper we investigate several problems concerning the existence and enumeration of highly irregular graphs as well as their independence numbers, with particular focus on the corresponding problems for highly irregular trees.
80 citations
01 Jan 1987
TL;DR: This paper estimates the size of C (k; n, p) and considers the probability that C(k) is k-connected when n + m, and estimates the maximal subgraph of G(n,p) with the minimal degree at least k = k(n).
Abstract: tuczak, T., Size and connectivity of the k-core of a random graph, Discrete Mathematics 91 (1991) 61-68. Let G(n,p) be a graph with n labelled vertices in which each edge is present independently with probability p =p(n) and let C(k; n, p) be the maximal subgraph of G(n,p) with the minimal degree at least k = k(n). In this paper we estimate the size of C(k; n, p) and consider the probability that C(k; n, p) is k-connected when n + m.
59 citations
TL;DR: This paper presents an O(ng+n log n) time algorithm for the dynamic connectivity problem in the case of the maximum genus of the derived graph being g and the best previously known time was Ω(n 2).
45 citations
TL;DR: The mutual exclusion problem is solved for an arbitrary connected graph and the solution is shown to be fair.
Abstract: We represent a set of communicating processes by a graph. The mutual exclusion problem is solved for an arbitrary connected graph. The solution is shown to be fair.
41 citations
01 Jan 1987
TL;DR: A number of interesting structural and algorithmic concepts from graph theory come together rather naturally, e.g., isometric embedding, Steiner points in graphs, retracts, diameters and linear programming.
Abstract: Suppose G is a fixed finite connected graph and for any two vertices x and y in G, dG(x,y) denotes the distance in G between x and y, i.e., the number of edges in a shortest path connecting x and y. Given an infinite sequence Q = (q1,q2,…) of vertices in G, suppose we would like to find another sequence P = PQ = (p0,p1,…) of vertices so that the quantity v ( P ) = lim sup n → ∞ 1 N ( ∑ i = 1 N ( d ( p i - 1 , p i ) + d ( p i , q i ) ) ) is as small as possible. This question represents a general formulation of a class of problems arising in self-adjusting data structures. In this paper we will investigate this and a number of related graph searching problems, such as requiring ***pn to be chosen before qn+i, i ≥ k, is known, and show how a number of interesting structural and algorithmic concepts from graph theory come together rather naturally, e.g., isometric embedding, Steiner points in graphs, retracts, diameters and linear programming.
32 citations
TL;DR: It is shown that an algorithm resilient to at most t faulty processors and k faulty links subject to omission or stopping faults exist, if and only if the network has a connectivity pair (t′, k′)>(t, k).
Abstract: We investigate the problem of reaching Byzantine Agreement in arbitrary networks where both processors and communication links are subject to omission or stopping faults For the case of deterministic, synchronous algorithms we give a necessary and sufficient condition relating the solvability of the problem to the connectivity of the network In particular, we show that an algorithm resilient to at mostt faulty processors andk faulty links subject to omission or stopping faults exist, if and only if the network has a connectivity pair (t′, k′)>(t, k)
27 citations
TL;DR: The analysis involves the surviving route graph, which consists of all nonfaulty nodes in the network with two nodes being connected by a directed edge iff the route from the first to the second is still intact after a set of component failures.
Abstract: We analyze the problem of constructing a network with a given number of nodes which has a fixed routing and which is highly fault tolerant. A construction is presented which forms a “product route graph” from two or more constituent “route graphs.” The analysis involves the surviving route graph , which consists of all nonfaulty nodes in the network with two nodes being connected by a directed edge iff the route from the first to the second is still intact after a set of component failures. The diameter of the surviving route graph is a measure of the worst-case performance degradation caused by the faults. The number of faults tolerated, the diameter, and the degree of the product graph are related in a simple way to the corresponding parameters of the constituent graphs. In addition, there is a “padding theorem” which allows one to add nodes to a graph and to extend a previous routing.
26 citations
TL;DR: Every edge-preserving self-map of a median graph leaves a cube invariant, which extends a fixed edge theorem for trees and parallels a result on invariant simplices in contractible graphs.
20 citations
TL;DR: A finite distance space X, d d: X2 → ℤ is hypermetric (of negative type) if ∑axayd(x, y) ≤ 0 for all integral sequences{ax∣x ∈ X} that sum to 1 (sum to 0).
Abstract: A finite distance spaceX, d d: X 2 ? ? is hypermetric (of negative type) if ?a x a y d(x, y) ≤ 0 for all integral sequences{a x ?x ? X} that sum to 1 (sum to 0).X, d is connected if the set {(x, y)?d(x, y) = 1, x, y ? X} is the edge set for a connected graph onX, and graphical ifd is the path length distance for this graph. Then we prove
18 citations
Journal Article•
TL;DR: In this paper, Carathtodory, Helly and Radon type theorems are proved for M-convex sets, and the Radon number is one more than the Helly number except possibly for triangle-free graphs, where it is at most 4.
17 citations
TL;DR: It is shown that for k ≥ 3, every k -connected graph G with girth at least 4 contains an induced cycle C such that G − V ( C ) is ( k − 2)-connected.
TL;DR: It is shown that the time-complexity of k-VCUAP (k≥2) is θ(k|V|) for every rooted directed tree.
TL;DR: The Euler genus of the surface Σ obtained from the sphere by the addition of k crosscaps and h handles is e ( Σ ) = k + 2 h and for a graph G, theEuler genus e ( G ) of G is the smallest Euler family among all surfaces in which G embeds.
TL;DR: It is shown that any two cartesian factorizations of a connected graph have a strict common refinement, improving on the unique factorization theorem of G. Sabidussi.
27 Apr 1987
TL;DR: In this paper, the concepts of dependency, connectivity, and reparability which arise in security verification are handled well by graph theory and are particularly useful for describing the hierarchical/modular design of a system.
Abstract: In designing and analyzing the structure and security of a large scale computer program or system, one is often interested in how it is constructed out of pieces or in the flow of information through it. The concepts of dependency, connectivity, and reparability which arise in security verification are handled well by graph theory. As is shown in this paper, graph homomorphism are particularly useful for describing the hierarchical/modular design of a system and for demonstrating the security of information flow.
TL;DR: It is shown that if a connected graph G admits a finite covering that has a k-factor, then there is a 1- or 2-fold covering of G that has been given as a canonical bipartite graph associated with G.
Abstract: It is shown that if a connected graph G admits a finite covering that has a k-factor, then there is a 1- or 2-fold covering of G that has a k-factor and that such a covering can be given as a canonical bipartite graph associated with G.
11 Aug 1987
TL;DR: Initial iconic alphanumeric data base results presented are most promising and the use of the basic concepts of such a directed graph processor offer significant impact on: associative, symbolic, inference, feature space and correlation-based AI processors, as well as on knowledge base organization and procedural knowledge control of AI processors.
Abstract: A directed graph processor and several optical realizations of its input symbolic feature vectors and the multi-processor operations required per node are given. This directed graph processor has advantages over tree and other hierarchical processors because of its large number of interconnections and its ability to adaptively add new nodes and restructure the graph. The use of the basic concepts of such a directed graph processor offer significant impact on: associative, symbolic, inference, feature space and correlation-based AI processors, as well as on knowledge base organization and procedural knowledge control of AI processors. Initial iconic alphanumeric data base results presented are most promising.