Showing papers on "Connectivity published in 1991"

Applications of a poset representation to edge connectivity and graph rigidity

Abstract: A poset representation for a family of sets defined by a labeling algorithm is investigated. Poset representations are given for the family of minimum cuts of a graph, and it is shown how to compute them quickly. The representations are the starting point for algorithms that increase the edge connectivity of a graph, from lambda to a given target tau = lambda + delta , adding the fewest edges possible. For undirected graphs the time bound is essentially the best-known bound to test tau -edge connectivity; for directed graphs the time bound is roughly a factor delta more. Also constructed are poset representations for the family of rigid subgraphs of a graph, when graphs model structures constructed from rigid bars. The link between these problems is that they all deal with graphic matroids. >

Constructing multidimensional spanner graphs

Abstract: Given a connected graph G=(V,E) with positive edge weights, define the distance dG(u,v) between vertices u and v to be the length of a shortest path from u to v in G. A spanning subgraph G' of G is said to be a t-spanner for G if, for every pair of vertices u and v, dG'(u,v)≤t·dG(u,v). Consider a complete graph G whose vertex set is a set of n points in and whose edge weights are given by the Lp distance between respective points. Given input parameter ∊, 0<∊≤1, we show how to construct a (1+∊)-spanner for G containing edges in time. We apply this spanner to the construction of approximate minimum spanning trees.

Reducing edge connectivity to vertex connectivity

Abstract: We show how to reduce edge connectivity to vertex connectivity. Using this reduction, we obtain a linear-time algorithm for deciding whether an undirected graph is 3-edge-connected, and for computing the 3-edge-connected components of an undirected graph.

The complexity of the residual node connectedness reliability problem

Abstract: This paper considers a probabilistic network in which the edges are perfectly reliable but the nodes fail with some known probabilities. The network is in an operational state if the surviving nodes induce a connected graph. The residual node connectedness reliability $R(G)$ of a network G is the probability that the graph induced by the surviving nodes is connected. This reliability measure is very different from the widely studied K-terminal network reliability measure. It is proven that the problem of computing the residual connectedness reliability is NP-hard by showing that the problem of counting the number of node induced connected subgraphs of a given graph is $# {\bf P}$-complete. The problem remains $# {\bf P}$-complete for split graphs as well as planar and bipartite graphs.

Exact graph-reduction algorithms for network reliability analysis

Abstract: Consideration is given to the problem of analyzing the reliability of a network, specifically, the probability that a given set of critical nodes within the network can communicate, given the failure probabilities for component nodes and links. An exact graph-reduction algorithm is presented for solving the k-terminal reliability problem with node failures on an arbitrary network. k-terminal reliability means that a specific set of k target nodes must be able to communicate with one another. The authors model the network by an undirected probabilistic graph whose vertices represent the nodes and whose edges represent the links. >

Computing the strength of a graph

Abstract: The strength of a graph is a measure of its vulnerability, strictly generalizing the edge connectivity of a weighted graph. Cunningham [J. Assoc. Comput. Mach., 32 (1985), pp. 549–562] showed how to compute the strength of a graph in $O(| V |^4 | E |)$ time, using ideas from polymatroids and network flow. In this paper, his polymatroid approach is modified, a modified version of the Goldberg-Tarjan network flow algorithm [J Assoc. Comput. Mach., 4 (1988), pp. 136-146] is used. Then, using ideas developed by Gallo, Grigoriadis, and Tarjan [SIAM J. Comput., 18 (1989), pp. 30–55], and by Gusfield, Martel, and Fernandez-Baca in [SIAM J. Comput.,16 (1987), pp. 237–251], it is shown that the solution runs in $O(| V |^3 | E |)$ time. Analogous speedups for sparse-case bounds are also obtainable.

Efficient parallel algorithms for testing k -connectivity and finding disjoint s-t paths in graphs

Abstract: An efficient parallel algorithm for testing whether a graph G is k-vertex connected is presented. The algorithm runs in $O(k^2 \log n)$ time and uses $(n + k^2 )kC(n,m)$ processors on a CROW PRAM, where n and m are the number of vertices and edges of G, and $C(n,m)$ is the number of processors required to compute the connected components of G in logarithmic time. For fixed k, the algorithm runs in logarithmic time and uses $nC(n,m)$ processors. To develop our algorithm, an efficient parallel algorithm is designed for the following disjoint s-t paths problem. Given a graph G, and two specified vertices s and t, find k vertex disjoint paths between s and t, if they exist. If no such paths exist, find a set of at most $k-1$ vertices whose removal disconnects s and t. The parallel algorithm for this problem runs in $O(k^2 \log n)$ time and uses $kC(n,m)$ processors. The way to modify the algorithm to find k-edge disjoint paths, if they exist, is shown. This yields an efficient parallel algorithm for testing w...

Tree-matchings in graph processes

Abstract: For a tree T a perfect T-matching in a graph G is a subgraph of G with at least $|G| - |T| + 1$ vertices, each component of which is isomorphic to T. Two properties, $\mathcal{A}$ and $\mathcal{B}$, are introduced where the former is a modification of the fact that the largest component of G has a perfect T-matching and the latter is a suitably chosen necessary condition for $\mathcal{A}$ expressed in terms of forbidden “pendant” subgraphs. We show that in the random graph process $\hat G_n $ the hitting times of both above properties coincide. This paper is the first one that deals with the hitting times of nonmonotone graph properties. It extends results of Bollobas and Frieze [Ann. Discrete Math., 28 (1985), pp. 23–46] and Bollobas and Thomason [Ann. Discrete Math., 28 (1985), pp. 47–98].

Do 3n−5 edges force a subdivision of K 5 ?

Abstract: : A conjecture of Dirac states that every simple graph with n vertices and 3n - 5 edges must contain a subdivision of K5. We prove that a topologically minimal counterexample is 5-connected, and that no minor-minimal counterexample contains K sub 4 - e. Consequently, we prove Dirac's conjecture for all graphs that can be imbedded in a surface with Euler characteristic at least -2.

Geodesics and bounded harmonic functions on infinite planar graphs

Abstract: It is shown there that an infinite connected planar graph with a uniform upper bound on vertex degree and rapidly decreasing Green's func- tion (relative to the simple random walk) has infinitely many pairwise finitely- intersecting geodesic rays starting at each vertex. We then demonstrate the existence of nonconstant bounded harmonic functions on the graph.

Graph Rewriting Systems and their Application to Network Reliability Analysis

Abstract: We propose a new kind of Graph Rewriting Systems (GRS) that provide a theoretical foundation for using the reduction methods to analyze network reliability, and give the critical pair lemma in this paper.

Topological analysis of satellite-based distributed sensor networks

Abstract: A method for evaluating the topological quality of networks for distributed sensor applications is presented. The criteria for evaluation are network survivability and delay. Nodal connectivity is used to characterize survivability, and mean path length is used to characterize delay. To calculate these two quantities, an algorithm is developed to find k shortest node-disjoint paths between a pair of nodes when each link has unit distance. The effects of nodal losses in a multiple-node satellite network consisting of a Walker low-orbit sphere and a geosynchronous constellation are examined. The example demonstrates how topological analysis on the basis of connectivity and mean path length may be used to detect, and subsequently to address, potential flaws in a network design. The results show that the geosynchronous/low-orbit link assignment protocol should be a primary concern in the design of this network. They also show that the nodal degree of a failed node, and the distribution of links between the Walker sphere and the geosynchronous constellation, are the fundamental determinants of mean path length. >

Applications of a Poset Representation to Edge Connectivity and Graph Rigidity ; CU-CS-545-91

Abstract: A poset representation for a family of sets defined by a labeling algorithm is investigated. Poset representations are given for the family of minimum cuts of a graph, and it is shown how to compute them quickly. The representations are the starting point for algorithms that increase the edge connectivity of a graph, from lambda to a given target tau = lambda + delta , adding the fewest edges possible. For undirected graphs the time bound is essentially the best-known bound to test tau -edge connectivity; for directed graphs the time bound is roughly a factor delta more. Also constructed are poset representations for the family of rigid subgraphs of a graph, when graphs model structures constructed from rigid bars. The link between these problems is that they all deal with graphic matroids.<>

Polyhedral embeddings in the projective plane

Abstract: We characterize the graphs that have polyhedral embeddings in the projective plane. We also prove that if one embedding of a graph is polyhedral then all embeddings of that graph are polyhedral.

Subgraphs, closures and hamiltonicity

Abstract: Closure theorems in hamiltonian graph theory are of the following type: Let G be a 2- connected graph and let u, v be two distinct nonadjacent vertices of G. If condition c(u,v) holds, then G is hamiltonian if and only if G + uv is hamiltonian. We discuss several results of this type in which u and v are vertices of a subgraph H of G on four vertices and c(u, v) is a condition on the neighborhoods of the vertices of H (in G). We also discuss corresponding sufficient conditions for hamiltonicity of G.

On the complexity of a fault-tolerance model for multicomputer systems

Abstract: In topological design of multicomputer systems (e.g., the hypercube), the edge- and vertex-connectivities have traditionally been used as deterministic measures of fault-tolerance. These measures have been noted to have some deficiencies and as a result several generalizations of graph connectivity have been proposed. In this paper, the authors examine some instances of the connectivity generalization proposed by Esfahanian and Hakimi, (1988). This generalization of graph connectivity can be used to model the fault-tolerance analysis of multicomputers in which any set S of the multicomputer components is considered fault free if the set S does not satisfy some given topological property rho . Using this model and different definitions of rho , the authors establish the complexity of analyzing the fault-tolerance of multicomputers. >

Paper: Parallel strong orientation on a mesh connected computer

Abstract: We present a solution for the following problem: given an undirected bridgeless connected graph G = (V, E), find an orientation of each edge such that the resulting directed graph is strongly connected. We assume the input graph to be given as an adjacency matrix stored in the processors of a mesh connected processor array such that each processor contains one entry. Our algorithm is a modification of Tarjan and Vishkin's strong orientation method [13] for a CREW PRAM. It takes time O(n) for an n-vertex graph which is asymptotically optimal for the considered computer model.