Showing papers on "Connectivity published in 1992"

A linear-time algorithm for finding a sparse k -connected spanning subgraph of a k -connected graph

Abstract: We show that anyk-connected graphG = (V, E) has a sparsek-connected spanning subgraphG′ = (V, E′) with ¦E′¦ =O(k¦V¦) by presenting anO(¦E¦)-time algorithm to find one such subgraph, where connectivity stands for either edge-connectivity or node-connectivity. By using this algorithm as preprocessing, the time complexities of some graph problems related to connectivity can be improved. For example, the current best time boundO(max{k 2¦V¦1/2,k¦V¦}¦E¦) to determine whether node-connectivityK(G) of a graphG = (V, E) is larger than a given integerk or not can be reduced toO(max{k 3¦V¦3/2,k 2¦V¦2}).

Sparsification-a technique for speeding up dynamic graph algorithms

Abstract: The authors provide data structures that maintain a graph as edges are inserted and deleted, and keep track of the following properties: minimum spanning forests, best swap, graph connectivity, and graph 2-edge-connectivity, in time O(n/sup 1/2/log(m/n)) per change; 3-edge-connectivity, in time O(n/sup 2/3/) per change; 4-edge-connectivity, in time O(n alpha (n)) per change; k-edge-connectivity, in time O(n log n) per change; bipartiteness, 2-vertex-connectivity, and 3-vertex-connectivity, in time O(n log(m/n)) per change; and 4-vertex-connectivity, in time O(n log(m/n)+n alpha (n)) per change. Further results speed up the insertion times to match the bounds of known partially dynamic algorithms. The algorithms are based on a technique that transforms algorithms for sparse graphs into ones that work on any graph, which they call sparsification. >

New sparseness results on graph spanners

Abstract: Let G=(V,E) be an n-vertex connected graph with positive edge weights. A subgraph G′ = (V,E′) is a t-spanner of G if for all u, v e V,the weighted distance between u and v in G′ is at most t times the weighted distance between u and v in G. We consider the problem of constructing sparse spanners, and the weight, defined as the sum of the edge weights in the spanner. In this paper, we concentrate on constructing spanners of small weight.For an arbitrary positive edge-weighted graph G, for any t> 1, and any e>0, we show that a t-spanner of G with weight O(n(2+e)/(t-1>) •wt(MST) can be constructed in polynomial time. We also show that (log2n)-spanners of weight O(log n)•wt(MST) can be constructed.We then consider spanners for complete graphs induced by a set of points in d-dimensional real normed space. The weight of an edge xy is the norm of the xy vector. We show that for these graphs, t-spanners with total weight O(log n)•wt(MST) can be constructed in polynomial time.

Maintaining bridge-connected and biconnected components on-line

Abstract: We consider the twin problems of maintaining the bridge-connected components and the biconnected components of a dynamic undirected graph. The allowed changes to the graph are vertex and edge insertions. We give an algorithm for each problem. With simple data structures, each algorithm runs inO(n logn +m) time, wheren is the number of vertices andm is the number of operations. We develop a modified version of the dynamic trees of Sleator and Tarjan that is suitable for efficient recursive algorithms, and use it to reduce the running time of the algorithms for both problems toO(mα(m,n)), where α is a functional inverse of Ackermann's function. This time bound is optimal. All of the algorithms useO(n) space.

The Complexity of Graph Connectivity

Abstract: In this paper we survey the major developments in understanding the complexity of the graph connectivity problem in several computational models, and highlight some challenging open problems.

Roots of the reliability polynomial

Abstract: The reliability of a graph G is the probability that G is connected, given that edges are independently operational with probability p. This is known to be a polynomial in p, and the location of the roots of these functions is discussed. In particular, it is conjectured that the roots of the reliability polynomial of any connected graph lie in the disc $| z - 1 | \leq 1$, and evidence for this conjecture is provided. It is shown that all real roots lie in $\{ 0 \} \cup ( 1,2 ]$ and that every graph has a subdivision for which the roots of the reliability polynomial lie in the conjectured disc.

On-line Steiner trees in the Euclidean plane

Abstract: Suppose we are given a sequence of n points v1,…,vn in the Euclidean plane, and our objective is to construct, on-line, a connected graph that connects all of them, trying to minimize the total sum of lengths of its edges. We assume that the points appear one at a time, vi arriving at step i. At the end of step i, the on-line algorithm must construct a connected graph Ti-1. This can be done by joining vi (not necessarily by a straight line) to any point of Ti-1, which need not necessarily be one of the previously given points vj. The performance of our algorithm is measured by its competitive ratio: the supremum, over all sequences v1,…,vn as above, of the ratio between the total length of the graph constructed by our algorithm and the total length of the best Steiner tree that connects all the points v1,…, vn. There are known on-line algorithms whose competitive ratio is O(log n), but there is no known nontrivial lower bound for the best possible competitive ratio. Here we prove that the upper bound is almost tight by establishing an O(log n/log log n) lower bound for the competitive ratio of any on-line algorithm. The lower bound holds for deterministic algorithms as well as for randomized ones, and obviously holds in any Euclidean space of dimension greater than 2 as well.

Cartesian graph factorization at logarithmic cost per edge

Abstract: LetG be a connected graph withn vertices andm edges. We develop an algorithm that finds the (unique) prime factors ofG with respect to the Cartesian product inO(m logn) time andO(m) space. This shows that factoringG is at most as costly as sorting its edges. The algorithm gains its efficiency and practicality from using only basic properties of product graphs and simple data structures.

Isoperimetric Inequalities and Transient Random Walks on Graphs

Abstract: recurrent. We show that any graph satisfying an isoperimetric inequality only slightly stronger than that of Z2 is transient. More precisely, if f(k) denotes the smallest number of vertices in the boundary of a connected subgraph with k vertices, then the graph is transient if the infinite sum E f(k)-2 converges. This can be applied to parabolicity versus hyperbolicity of surfaces. 1. Introduction. Let G be a connected graph which is locally finite, that is, all vertices have finite degree. We consider a random walk starting at a vertex v, say, such that at any vertex u, the walk proceeds to a neighbour with probability 1/d(u), where d(u) is the degree (i.e., the number of neighbours) of u. The graph G is recurrent if we revisit v with probability 1. Otherwise G is transient. It is well known that the three-dimensional grid Z3 is transient while Z2 is recurrent. More generally, Nash-Williams [13] (see also [4, 10]) proved that any graph with smaller growth rate than Z2 is recurrent. Lyons [10] showed that certain subgraphs of grids are transient provided they grow just a little faster than Z2. Other results, in terms of isoperimetric inequalities, supporting the statement that Z2 is, in a sense, an "extreme" recurrent graph, can be derived from work of Fernandez [5], Grigor'yan [7] and Varopoulos [15]. (Varopoulos [16] used results of Gromov [8] to characterize completely the recurrent Cayley graphs.) We shall carry these results further. If V is a vertex set in G, then aV will denote the boundary of V, that is, the set of vertices of V having neighbours outside of V. Let f be a nondecreasing positive real function defined on the natural numbers. We say that G satisfies an f-isoperimetric inequality if there exists a constant c > 0 such that, for each finite vertex set V of G,

Symbolic and geometric connectivity graph methods for route planning in digitized maps

Abstract: The results of research involving spatial reasoning within digitized maps are reported, focusing on techniques for 2D route planning in the presence of obstacles. Two alternative approaches to route planning are discussed, one involving heuristic symbolic processing and the other employing geometric calculations. Both techniques employ A* search over a connectivity graph. The geometric system produces a simple list of coordinate positions, whereas the symbolic system generates a symbolic description of the planned route. The symbolic system achieves this capability through the use of inference rules that can analyze and classify spatial relationships within the connectivity graph. The geometric method calculates an exact path from the connectivity information in the graph. Thus, the connectivity graph acts both as a knowledge structure on which spatial reasoning can be performed and as a data structure supporting geometrical calculations. An extension of the methodology that exploits a hierarchical data structure is described. >

Cayley graphs with optimal fault tolerance

Abstract: If H is a quasiminimal generating set for a finite group G, it is proved that the Cayley graph (G; H) has optimal fault tolerance unless it belongs to a special family. >

Discrete Schrödinger operators on a graph

Abstract: In this paper, we study some spectral properties of the discrete Schrodinger operator = Δ + q defined on a locally finite connected graph with an automorphism group whose orbit space is a finite graph. The discrete Laplacian and its generalization have been explored from many different viewpoints (for instance, see [2] [4]). Our paper discusses the discrete analogue of the results on the bottom of the spectrum established by T. Kobayashi, K. Ono and T. Sunada [3] in the Riemannian-manifold-setting.

Fully dynamic algorithms for 2-edge connectivity

Abstract: This paper studies the problem of maintaining the 2-edge-connected components of a graph undergoing repeated dynamic modifications, such as edge insertions and edge deletions. It is shown how to test at any time whether two vertices belong to the same 2-edge-connected component, and how to insert and delete an edge in $O(m^{2/3} )$ time in the worst case, where m is the current number of edges in the graph. This answers a question posed by Westbrook and Tarjan [Tech. Report CS-TR-229-89, Dept. of Computer Science, Princeton University, Princeton, NJ, August 1989; Algorithmica., to appear].For planar graphs, the paper presents algorithms that support all these operations in $O(\sqrt {n\log \log n} )$ worst-case time each, where n is the total number of vertices in the graph.

Directed s-t numberings, rubber bands, and testing digraph k-vertex connectivity

Abstract: Let G = (V, E) be a directed graph and n denote |V|. We show that G is k-vertex connected iff for every subset X of V with |X| = k, there is an embedding of G in the (k-1)-dimensional space Rk-1, f : V →Rk-1, such that no hyperplane contains k points of {f(v) | v e V}, and for each v e V - X, f(v) is in the convex hull of {f(w) | (v, w) e E}. This result generalizes to directed graphs the notion of convex embeddings of undirected graphs introduced by Linial, Lova´sz and Wigderson in “Rubber bands, convex embeddings and graph connectivity,” Combinatorica 8 (1988), 91-102. Using this characterization, a directed graph can be tested for k-vertex connectivity by a Monte Carlo algorithm in time O((M(n) + nM(k)).(log n)) with error probability n, and by a Las Vegas algorithm in expected time O((M(n)+nM(k)).k), where M(n) denotes the number of arithmetic steps for multiplying two n x n matrices (M(n) = O(n2.3755)). Our Monte Carlo algorithm improves on the best previous deterministic and randomized time complexities for k > n0.19; e.g., for k = (n0.5, the factor of improvement is > n0.62. Both algorithms have processor efficient parallel versions that run in O((log n)2) time on the EREW PRAM model of computation, using a number of processors equal to (log n) times the respective sequential time complexities. Our Monte Carlo parallel algorithm improves on the number of processors used by the best previous (Monte Carlo) parallel algorithm by a factor of at least (n2/(log n)3) while having the same running time. Generalizing the notion of s-t numberings, we give a combinatorial construction of a directed s-t numbering for any 2-vertex connected directed graph.

Polyhedra of the equivalent subgraph problem and some edge connectivity problems

Abstract: In this paper the problem of finding a minimum weight equivalent subgraph of a directed graph is considered. The associated equivalent subgraph polyhedron $P ( G )$ is studied. Several families of facet-defining inequalities are described for this polyhedron. A related problem of designing networks that satisfy certain survivability conditions, as introduced in [M. Grotschel and C. L. Monma, SIAM Journal on Discrete Mathematics, 3 (1990), pp. 502–523] is also studied. The low connectivity case is formulated on directed graphs, and the directed formulation is shown to give a better LP-relaxation than the undirected one. It is shown how facet-defining inequalities of $P ( G )$ give facet-defining inequalities in this case. Computational results are presented for some randomly generated problems.

Scattering Matrices for Micro Schemes

Abstract: A mathematical model for a simple microscheme is constructed on the basis of the scattering theory for a pair of different self-adjoint extensions of the same symmetric ordinary differential operator on a one-dimensional manifold, which consist of a finite number of semiinfinite straight outer lines attached to a “black box” in a form of a flat connected graph. An explicit expression for the scattering is given under a continuity condition at the graph vertices.

Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity

Abstract: Two Sequences of 2-Regular Graceful Graphs Consisting of 4-gons (J. Abrham, A. Kotzig#). A Survey of Self-Dual Polyhedra (D. Archdeacon). On Magic Labellings of Convex Polytopes (M. Baca). A Packing Problem and Geometrical Series (V. Balint). On the Bananas Surface B2 (R. Bodendiek, K. Wagner). Structural Properties and Colorings of Plane Graphs (O.V. Borodin). The Binding Number of Graphs (M. Borowiecki). Note on Algorithmic Solvability of Trahtenbrot-Zykov Problem (P. Bugata). Cartesian Dimensions of a Graph (G. Burosch, P. V. Ceccherini). The Steiner Minimal Tree Problem in L2p (D. Cieslik). On k -Connected Subgraphs of the Hypercube (T. Dvorak). On Some of My Favourite Problems in Various Branches of Combinatorics (P. Erdos). Realizability of Some Starlike Trees (D. Froncek). The Construction of All Configurations (124, 163) (H. Gropp). (p,q)-realizability of Integer Sequences with Respect to Mobius Strip (M. Hornak). Vertex Location Problems (O. Hudec). On Generation of a Class of Flowgraphs (A.J.C. Hurkens, C.A.J. Hurkens, R.W. Whitty). The Weight of a Graph (J. Ivanco). On the Kauffman Polynomial of Planar Matroids (F. Jaeger). On Symmetry Groups of Selfdual Convex Polyhedra (S. Jendrol). A Remark on 2- (u,kappa,lambda) Designs (V. Jurak). On a New Class of Intersection Graphs (M. Koebe). Asymptotic Normality of Isolated Edges in Random Subgraphs of the n -Cube (U. Konieczna). On Bounds of the Bisection Width of Cubic Graphs (A.V. Kostochka, L.S. Mel'nikov). On Random Cubical Graphs (A.V. Kostochka, A.A. Sapozhenko, K. Weber). On the Computational Complexity of Seidel's Switching (J. Kratochvil, J. Nesetril, O. Zyka). The Harmonious Chromatic Number of a Graph (A. Kundrik). Arboricity and Star Arboricity of Graphs (A. Kurek). Extended 4-Profiles of Hadamard Matrices (C. Lin, W.D. Wallis, Z. Lie). Good Family Packing (M. Loebl, S. Poljak). Solution of an Extremal Problem Concerning Edge-Partitions of Graphs (Z. Lonc). Balanced Extensions of Spare Graphs (T. Luczak, A. Rucinski). Two Results on Antisocial Families of Balls (A. Malnic, B. Mohar). Hamiltonicity of Vertex-transitive pq -Graphs (D. Marusic). On Nodes of Given Out-Degree in Random Trees (A. Meir, J.W. Moon). All Leaves and Excesses are Realizable for k =3 and All lambda (E. Mendelsohn, N. Shalaby, S. Hao). The Binding Number of k -Trees (D. Michalak). An Extension of Brook's Theorem (P. Mihok). On Sectors in a Connected Graph (L. Nebesky). Irreconstructability of Finite Undirected Graphs from Large Subgraphs (V. Nydl). On Inefficient Proofs of Existence and Complexity Classes (C.H. Papadimitriou). Optimal Coteries on a Network (C.H. Papadimitriou, M. Sideri). On Some Heuristics for the Steiner Problem in Graphs (J. Plesnik). Minimax Results and Polynomial Algorithms in VLSI Routing (A. Recski). Critical Perfect Systems of Difference Sets (D.G. Rogers). Some Operations (Not) Preserving the Integer Rounding Property (A.

On the graph bisection problem

Abstract: Some interesting theoretical aspects of graph bisection, a fundamental problem with several applications in the design of very-large-scale-integrated (VLSI) circuits, are presented. Sufficient conditions of optimality are presented, along with a bound on the cost of an optimal bisection. It is then shown that for dense graphs a bisection that approximates an optimal one can be easily found by using a simple greedy method. A class of graphs for which the ratio of an upper and a lower bound on the optimal cost approaches one as the number of vertices in the graph increases is exhibited. >

On alternative p-center problems

Abstract: LetG=(V, E) be an undirected connected graph with positive edge lengths. The vertexp-center problem is to find the optimal location ofp centers so that the maximum distance to a vertex from its nearest center is minimized, where the centers may be placed at the vertices. Kariv and Hakimi have shown that this problem is NP-hard. We will consider two modifications of this problem in which each center may be located in one of two predetermined vertices. We will show the NP-completeness of their recognition versions.

Two algorithms for the subset interconnection design problem

Abstract: An NP-complete generalization of the minimum spanning tree problem is considered. Given are a set of V, a cost function c: V × V R+, and a collection {X1, …, Xm} of subsets of V. A graph G with vertex set V is called feasible if every Xi induces a connected subgraph of G. The minimum subset interconnection design problem is to find a feasible graph with a minimum cost sum. In this paper, two heuristic algorithms for the problem are given and analyzed. Several classes of input data for which one of the algorithms finds optimal or at least approximative solutions are presented.