Showing papers on "Connectivity published in 1988"
TL;DR: Lower and upper bounds for λ( G : P ) when P is defined as follows: A graph H satisfies property P if it contains more than one vertex and a new generalization of the notion of connectivity is given.
331 citations
TL;DR: A communication protocol is described which uses k rooted spanning trees having the property that for every vertex ν the paths from ν to the root are edge-disjoint, and an algorithm to find two such trees in a 2-edge connected graph that runs in time proportional in the number of edges in the graph.
Abstract: Consider a network of asynchronous processors communicating by sending messages over unreliable lines. There are many advantages to restricting all communications to a spanning tree. To overcome the possible failure of k′ < k edges, we describe a communication protocol which uses k rooted spanning trees having the property that for every vertex ν the paths from ν to the root are edge-disjoint. An algorithm to find two such trees in a 2-edge connected graph is described that runs in time proportional in the number of edges in the graph. This algorithm has a distributed version which finds the two trees even when a single edge fails during their construction. The two trees then may be used to transform certain centralized algorithms to distributed, reliable, and efficient ones.
297 citations
24 Oct 1988
TL;DR: A general framework for the study of a broad class of communication problems is developed based on a recent analysis of the communication complexity of graph connectivity, which makes use of combinatorial lattice theory.
Abstract: A general framework for the study of a broad class of communication problems is developed. It is based on a recent analysis of the communication complexity of graph connectivity. The approach makes use of combinatorial lattice theory. >
135 citations
TL;DR: In this paper, the spectral radius ϱ(A) of a simple connected graph with n vertices and m edges is defined as the adjacency matrix of the graph G and A is a matrix whose spectral radius is equal to 2m − n + 1 with equality if and only if G is isomorphic to one of the following two graphs: (a) the star K1,n−1; (b) the complete graph Kn.
129 citations
TL;DR: Probabilistic (Monte-Carlo and Las Vegas) algorithms for computing the connectivity of a graph and their algorithms are faster than the best known (deterministic) connectivity algorithms for allk≧√n, and for very dense graphs the Monte Carlo algorithm is faster by a linear factor.
Abstract: We give various characterizations ofk-vertex connected graphs by geometric, algebraic, and “physical” properties. As an example, a graphG isk-connected if and only if, specifying anyk vertices ofG, the vertices ofG can be represented by points of ℝk−1 so that nok are on a hyper-plane and each vertex is in the convex hull of its neighbors, except for thek specified vertices. The proof of this theorem appeals to physics. The embedding is found by letting the edges of the graph behave like ideal springs and letting its vertices settle in equilibrium. As an algorithmic application of our results we give probabilistic (Monte-Carlo and Las Vegas) algorithms for computing the connectivity of a graph. Our algorithms are faster than the best known (deterministic) connectivity algorithms for allk≧√n, and for very dense graphs the Monte Carlo algorithm is faster by a linear factor.
100 citations
TL;DR: It is proved that every nontrivial bridged graph has a pair of adjacent vertices u, v, with v adjacent to everything to which u is adjacent, and every bridgedgraph G has a vertex u such that G − u is an isometric subgraph, and hence also bridged.
73 citations
TL;DR: In this paper, an undirected connected graph G with n nodes is considered and a subset F of nodes of G is a feedback vertex set (fvs) if G − F is a forest and if no two nodes of J are adjacent and G − J is connected.
Abstract: Let G be an undirected connected graph with n nodes. A subset F of nodes of G is a feedback vertex set (fvs) if G − F is a forest and a subset J of nodes of G is a nonseparating independent set (nsis) if no two nodes of J are adjacent and G − J is connected. f(G), z(G) denote the cardinalities of a minimum fvs and a maximum nsis, respectively, of G. The equation f(G) = n/2 − z(G) + 1 and two new upper bounds on f(G) are derived for cubic graphs G.
56 citations
TL;DR: In this paper, the connectivity of infinite random graphs is investigated and necessary and sufficient conditions for a zero-one law to hold are given when the edges are chosen independently, and specific examples are treated including one where the vertex set is N and the probability that an edge joiningi toj is present depends only on |i−j|.
Abstract: Several results concerning the connectivity of infinite random graphs are considered. A necessary sufficient condition for a zero–one law to hold is given when the edges are chosen independently. Some specific examples are treated including one where the vertex set isN and the probability that an edge joiningi toj is present depends only on |i−j|.
39 citations
TL;DR: An upper bound on the n-extendability of a graph in terms of its genus is obtained in this paper.
35 citations
01 Jan 1988
TL;DR: A general framework for the study of a broad class of communication problems which has several interesting special cases including the graph connectivity problem is developed, based on combinatorial lattice theory.
Abstract: In a recent paper, Hajnal, Maass and Tura'n analyzed the communication complexity of graph connectivity. Building on this work, we develop a general framework for the study of a broad class of communication problems which has several interesting special cases including the graph connectivity problem. The approach is based on combinatorial lattice theory.
27 citations
TL;DR: This paper proposes anO(|V|) time algorithm to improve Pawagi's result on the maximum weight independent set problem for a general graph and is time optimal within a constant factor.
Abstract: The maximum weight independent set problem for a general graph is NP-hard. But for some special classes of graphs, polynomial time algorithms do exist for solving it. Based on the divide-and-conquer strategy, Pawagi has presented anO(|V|log|V|) time algorithm for solving this problem on a tree. In this paper, we propose anO(|V|) time algorithm to improve Pawagi's result. The proposed algorithm is based on the dynamic programming strategy and is time optimal within a constant factor.
07 Jun 1988
TL;DR: The authors characterize a smallest 3-connectivity augmentation, involving the determination of a minimum set of edges to be added so as to3-connect a given undirected simple graph, and it is shown that there is an O- algorithm for finding a solution to the problem.
Abstract: The authors characterize a smallest 3-connectivity augmentation, involving the determination of a minimum set of edges to be added so as to 3-connect a given undirected simple graph. It is shown that there is an O( mod V mod ( mod V mod + mod E mod )/sup 2/) algorithm for finding a solution to the problem. >
TL;DR: The value ν (G) of this pursuit-evasion game for various classes of graphs G, as well as those optimal mixed strategies for achieving ν(G), is investigated.
Abstract: Two players, Red and Blue, each independently choose a vertex of a connected graph G. Red must then pay Blue an amount equal to the distance between the vertices chosen. In this note, we investigate the value ν(G)of this pursuit-evasion game for various classes of graphs G, as well as those optimal mixed strategies for achieving ν(G). It is shown that some rather counterintuitive behavior can occur. For example, there exist graphs G in which, for any optimal mixed strategy, Red never selects a vertex in the center of G.
TL;DR: In this article, the necessary and sufficient conditions for graphs to have a Δλ-cycle or Δλpath were derived and the results are analogues of the results of Chvatal and Erdos on Hamilton cycles and Hamiiton paths.
Abstract: A Cycle C of a graph G is called a Dλ-cycle if every component of G - V(C) has order less than λ A Dλ-path is defined analogously. Dλ-cycles and Dλ-paths were introduced by Veldman. Here a cycle C of a graph G is called a Δλ-cycle if all vertices of G are at distance less than λ from a vertex of C. A Δλ-path is defined analogously. In particular, in a connected graph, a Δλ-cycle is a Δλ-Cycle and a Δλ-Path is a Δλ-path. Furthermore, a Δ1-cycle is a Hamilton cycle and a Δ1path is a Hamilton path. Necessary conditions and sufficient conditions are derived for graphs to have a Δλ-cycle or Δλ-path. The results are analogues of theorems on Dλ-cycles and Dλ-paths. In particular, a result of Chvatal and Erdos on Hamilton cycles and Hamiiton paths is generalized. A recent conjecture of Bondy and Fan is settled.
TL;DR: The connectivity function k of a matroid M on a set E is defined by k(X) = r(X + r(E − X) − r (E) (X ⊆ E), where r is the rank function.
TL;DR: An ambiguity in the recently introduced concept of internal connectivity is pointed out and an alternative approach to solve this ambiguity is suggested.
TL;DR: In this article, the notion of local matrix with respect to a directed graph was introduced and it was shown that every matrix over the real or complex numbers can be represented by a local matrix.
Abstract: Let $D = ( V,E )$ be a directed graph with n vertices. We define the notion of a local matrix with respect to D and we show that every $n \times n$ matrix, over the real or complex numbers, can be ...
TL;DR: It is shown here that a connected graph G without subgraphs isomorphic to K4 is triangulated if and only if its chromatic polynomial P(G,λ) equals λ(λ − 1)m( λ − 2)r for some integers m ≧ 1, r ≧ 0.
Abstract: It is shown here that a connected graph G without subgraphs isomorphic to K4 is triangulated if and only if its chromatic polynomial P(G,λ) equals λ(λ − 1)m(λ − 2)r for some integers m ≧ 1, r ≧ 0. This result generalizes the characterization of Two-Trees given by E.G. Whitehead [“Chromaticity of Two-Trees,” Journal of Graph Theory9 (1985) 279–284].
01 Jan 1988
TL;DR: It is shown that the proposed systolic algorithm for finding bridges in an n-node connected graph is an improvement over the previously known algorithm on a n × n array of processing elements, as proposed by Attalah and Kosaraju.
Abstract: A new systolic algorithm for finding bridges in an n-node connected graph is proposed in this paper. The algorithm is executable on a n × n array of processing elements. It is shown that our algorithm is an improvement over the previously known algorithm for an n-node connected graph on a n × n array of processing elements, as proposed by Attalah and Kosaraju. The improvement both in area and time complexity is significant. Moreover, the control structure required for implementing the proposed algorithm on a systolic array is simpler.
TL;DR: It is shown that the connectivity of the d-ary De Bruijn graph is at least d − 1 and this bound is tight since the outdegree of this graph with self-loops deleted is d −1.
TL;DR: This paper proves the following generalizations of Katerinis' theorem: If G − x has a 2k-factor for each x ϵ V(G), then G has a2k-Factor.
12 Jun 1988
TL;DR: The author introduces a parameter known as the reliability domination of a graph and shows its role in calculating the reliability of the network reliability model.
Abstract: The author surveys some of the basic analysis and synthesis results in network reliability theory. The model considered is an undirected simple graph with independent probabilities that edges operate. The reliability is defined as the probability that the network has some path between each pair of nodes. The analysis problem is to calculate the reliability, given the graph and the probabilities. The author introduces a parameter known as the reliability domination of a graph and shows its role in calculating the reliability. A number of interesting synthesis problems concerning the reliability model are illustrated for the case in which the edge probabilities are all equal. Two specific synthesis problems are discussed in detail, giving the known results and the outstanding questions. The author shows how the topology of the optimal network might change as a function of the edge probabilities. >
01 Sep 1988
TL;DR: A frame control strategy is presented which directs an efficient graph search when information available for finding a path is distributed throughout the graph.
Abstract: A frame control strategy is presented which directs an efficient graph search when information available for finding a path is distributed throughout the graph. The frame structure requires (worst case) only O(K/sup N/) memory size instead of O(K/sup 2N/), which is needed for a matrix representation of an N-level, K-branch tree graph. An efficiency analysis is detailed, and its application to a communication system for the nonverbal, severely motor disabled is discussed. >
TL;DR: It is proved that if G is a connected graph with p vertices and minimum degree greater than max (p/4 − 1-3) then G2 is pancyclic and the result is best possible of its kind.
Abstract: We prove that if G is a connected graph with p vertices and minimum degree greater than max (p/4 − 1,3) then G2 is pancyclic. The result is best possible of its kind.
01 Aug 1988
TL;DR: The purpose of this paper is to present two analytical performance models which have been designed to understand the load-throughput behavior of Noahnet and show that the average number of successful messages in a slot given k partitions of transmissions is (M - k )/( N - 1).
Abstract: Noahnet is an experimental flood local area network with features such as high reliability and high performance. Noahnet uses a randomly connected graph topology with four to five interconnections per node and a flooding protocol to route messages.The purpose of this paper is to present two analytical performance models which we have designed to understand the load-throughput behavior of Noahnet. Both models assume slotted Noahnet operation and also assume that if k messages attempt transmission in a slot, the network gets divided into k partitions of arbitrary sizes - one partition for each message. First, we show that the average number of successful messages in a slot given k partitions of transmissions is (M - k)/(N - 1), where N is the number of nodes in the network and M is the number of nodes out of N that participate in the flooding of k messages. This is an interesting result and is used in both models to derive the load-throughput equations.Each model is then presented using a set of assumptions, derivations of load-throughput equations, a set of plots, and the discussion of results. Models one and two differ in the way they account for retransmissions.
TL;DR: A sufficient condition is established for the connected edge-cutset connectivity of a graph to equal its minimum degree that does not imply its edge-connectivity equals its minimumdegree.
15 Jun 1988
TL;DR: A procedure is described which given a Boolean equation of the AOI-type will generate an area-efficient layout topology for an electronic circuit implementing this function in MOS technology.
Abstract: A procedure is described which given a Boolean equation of the AOI-type will generate an area-efficient layout topology for an electronic circuit implementing this function in MOS technology n- or p-Transistors are assumed to be arranged in two horizontal rows with their gates formed by vertical polysilicon wires crossing the cell The topology generation problem essentially consists in interconnecting these transistors using metal wires The proposed procedure makes use of several graphs to reflect transistor network connectivity, potential conflicts between metal wires, potential embrace situations among them, and sharing of tracks between them Classical graph algorithms, such as colouring and critical path analysis, serve to resolve the conflicts and to assign every wire a legal location
01 Aug 1988
TL;DR: This report considers several problems concerning vertex connectivity of undirected graphs and presents new bounds and algorithms for these problems, which help in the design of optimum communication networks.
Abstract: : This report considers several problems concerning vertex connectivity of undirected graphs and presents new bounds and algorithms for these problems. Connectivity is one of the fundamental graph properties, and there has been a considerable amount of work on algorithms and structural aspects of this property. Applications of graph connectivity arise in operation research for scheduling problems, network analysis in electrical engineering, and many other real-life problems. The most direct application of this problem is for the reliability of networks. A fundamental criterion for evaluating performance of a communications network is its ability to withstand the failure of is not only real-life models but nonexistence of good upper bounds for the number of minimum size separating vertex sets of graphs. Another important measure of network reliability is to determine the subgraphs which are highly connected and to decompose the network into them. The results in all of these measures help in the design of optimum communication networks.
TL;DR: An intermediate value theorem is proved for k(B( )) and the minimum and maximum number of edges possible in a graph G of order n is investigated and the case when G is complete, so that is a tournament.