Showing papers on "Connectivity published in 1993"

Isoperimetry, logarithmic sobolev inequalities on the discrete cube, and margulis' graph connectivity theorem

Abstract: We prove that the suitably defined surface area of a subsetA of the cube {0,1}n is bounded below by a certain explicit function of the size ofA. We establish a family of logarithmic Sobolev inequalities on the cube related to this isoperimetric result. We also give a quantitative version of Margulis' graph connectivity theorem.

An efficient algorithm to locate all the load flow solutions

Abstract: A novel, numerically efficient algorithm to find all the solutions to load flow equations is presented. This algorithm is based on the analysis of the topological structures of the solution set defined by the parameterized load flow equation. It is found that the load flow solutions are connected by one-dimension manifolds (curves) to form a connected graph. All the load flow solutions can be found by tracing these curves. The significance of this algorithm is that it can be guaranteed to find all the solutions while reducing the amount of computations to the point that this task can be attempted on real power systems. Results on two example systems are provided to demonstrate the numerical performance of the algorithm and to provide a comparison with the currently available method. >

Communication complexity and combinatorial lattice theory

Abstract: In a recent paper, Hajnal, Maass, and Turan 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 the combinatorial theory of alignments and lattices.

Detour Distance in Graphs

Abstract: For vertices u and v in a connected graph G, the detour distance d* (u, v) between u and v is the length of a longest path P for which the subgraph induced by the vertices of P is P itself. A graph G is called a detour graph if d* (u, v) equals the standard distance between u and v in G for every pair u, v of vertices of G. Several results concerning detour distance and detour graphs are presented.

On-line steiner trees in the Euclidean plane

Abstract: Suppose we are given a sequence ofn points 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. The points appear one at a time, and at each step the on-line algorithm must construct a connected graph that contains all current points by connecting the new point to the previously constructed graph. This can be done by joining the new point (not necessarily by a straight line) to any point of the previous graph (not necessarily one of the given points). The performance of our algorithm is measured by its competitive ratio: the supremum, over all sequences of points, 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. There are known on-line algorithms whose competitive ratio isO(logn) even for all metric spaces, but the only lower bound known is of [IW] for some contrived discrete metric space. Moreover, for the plane, on-line algorithms could have been more powerful and achieve a better competitive ratio, and no nontrivial lower bounds for the best possible competitive ratio were known. Here we prove an almost tight lower bound of Ω(logn/log logn) 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.

On sparse subgraphs preserving connectivity properties

Abstract: From a theorem of W. Mader [“Uber minimal n-fach zusammenhangende unendliche Graphen und ein Extremal problem,” Arch. Mat., Vol. 23 (1972), pp. 553–560] it follows that a k-connected (k-edge-connected) graph G = (V,E) always contains a k-connected (k-edge-connected) subgraph G′ = (V,E′) with O(k|V|) edges. T. Nishizeki and S. Poljak “K-Connectivity and Decomposition of Graphs into Forests,” Discrete Applied Mathematics, submitted) showed how G′ can be constructed as the union of k forests. H. Nagamochi and T. Ibaraki [A Linear Time Algorithm for Finding a Sparse k-Connected Spanning Subgraph of a k-Connected Graph, Algorithmica, Vol. 7 (1992), pp. 583–596] constructed such a subgraph Gk in linear time and showed for any pair x,y of nodes that Gk contains k openly disjoint (edge-disjoint) paths connecting x and y if G contains k openly disjoint (edge-disjoint) paths connecting x and y (even if G is not k-connected (k-edge-connected)). In this article we provide a much shorter proof of a common generalization of the edge- and node-connectivity versions showing that the subgraph Gk has a certain mixed connectivity property. © 1993 John Wiley & Sons, Inc.

Delta-wye transformations and the efficient reduction of two-terminal planar graphs

Abstract: A simple, O(|V|2) time algorithm is presented that reduces a connected two-terminal, undirected, planar graph to a single edge, by way of series and parallel reductions and delta-wye transformations. The method is applied to a class of optimization/equilibnum problems which includes max flow, shortest path, and electrical resistance problems.

Complexity of the forwarding index problem

Abstract: The forwarding index problem is, given a connected graph G and an integer k, finding a way of connecting each ordered pair of vertices by a path so that every vertex is an internal point of at most k such paths. Such a problem arises in the design of communication networks and parallel architectures, a model of parallel computation being represented by a network of processors or machines processing and forwarding(synchronous) messages to each other and a physical constraint on the number of messages that can be processed by a single machine. In this paper, the author proves that the forwarding index problem is NP-complete even if the diameter of the graph is 2, thereby answering a question of F. Chung et al. [IEEE Trans. Inform. Theory, 33 (1987), pp. 224–232] concerning the complexity of the problem.

On the "log rank"-conjecture in communication complexity

Abstract: We show the existence of a non-constant gap between the communication complexity of a function and the logarithm of the rank of its input matrix. We consider the following problem: each of two players gets a perfect matching between two n-element sets of vertices. Their goal is to decide whether or not the union of the two matchings forms a Hamiltonian cycle. We prove: (1) The rank of the input matrix over the reals for this problem is 2/sup O(n)/. (2) The non-deterministic communication complexity of the problem is /spl Omega/(n log log n). Our result also supplies a superpolynomial gap between the chromatic number of a graph and the rank of its adjacency matrix. Another conclusion from the second result is an /spl Omega/(n log log n). Lower bound for the graph connectivity problem in the non-deterministic case. We make use of the theory of group representations for the first result. The second result is proved by an information theoretic argument. >

Time Space Tradeoffs (Getting Closer to the Barrier

Abstract: We survey the current status of the time-space complexity of various basic computational problems. For time-space (or space alone) lower bounds, Boolean branching programs are the “ultimate model”. We consider restricted and unrestricted branching program models as well as certain “structured models” which are appropriate to the particular problems being studied. Recent results on graph connectivity are especially noteworthy.

Space bounds for graph connectivity problems on node-named JAGs and node-ordered JAGs

Abstract: Two new models, NO-JAG and NN-JAG in order of increasing computation power, are introduced as extensions to the conventional JAG model. A space lower bound of /spl Omega/(log/sup 2/ n/log log n) is proved for the problem of directed st-connectivity on a probabilistic NN-JAG and a space upper bound of O(log n) is proved for the problem of directed st-nonconnectivity on a nondeterministic NO-JAG. It is also shown that a nondeterministic NO-JAG is nearly as powerful as a nondeterministic Turing machine. >

The independent domination number of a cubic 3-connected graph can be much larger than its domination number

Abstract: We show that the difference between independent domination number of a cubic 3-connected graph and its domination number can be arbitrarily large. This disproves a conjecture posed inGraphs and Combinatorics by C. Barefoot, F. Harary, and K.F. Jones [2].

Random sampling in matroids, with applications to graph connectivity and minimum spanning trees

Abstract: Random sampling is a powerful way to gather information about a group by considering only a small part of it. We give a paradigm for applying this technique to optimization problems, and demonstrate its effectiveness on matroids. Matroids abstractly model many optimization problems that can be solved by greedy methods, such as the minimum spanning tree (MST) problem. Our results have several applications. We give an algorithm that uses simple data structures to construct an MST in O(m+n log n) time. We give bounds on the connectivity (minimum cut) of a graph suffering random edge failures. We give fast algorithms for packing matroid bases, with particular attention to packing spanning trees in graphs. >

An integer polytope related to the design of survivable communication networks

Abstract: The problem of designing communication networks that can survive the loss of any single link is studied. Such problems can be formulated as minimum cost 2-edge connected subgraph problems in a complete graph. The linear programming cutting plane approach has been used effectively for related problems in [Schwerpunktprogramm der Deutschen Forschungsgemeinschaft, Anwendungsbezogene Optimierung and Steuerung, Report No. 188, 1989], where problem-specific cutting planes that define facets of the underlying integer polyhedra are used. This paper introduces a new class of valid inequalities for the polytope associated with the minimum cost 2-edge connected subgraph problem, and necessary and sufficient conditions for these inequalities to be facet-inducing for this polytope are given.

Method and apparatus for schematic routing

Abstract: An apparatus and method for generating connected graphs for display on a computer monitor. A data structure and virtual map are defined which provide a linked list of objects appearing at any location in the connected graph, the virtual map being larger than the dimensions of the display monitor if the connected graph demands larger dimensions. Commands carried out with respect to any object on the monitor are correlated with the cell maps in the virtual map, which point to the linked lists of objects, the lists in turn pointing to objects stored in an object records database. This provides rapid access to all objects at any given location on the display or in the connected graph. When a user gives a command to create a connection between two objects in the graph, the method generates the shortest connection possible, in terms of both pixel length and arc length for arc-connections, balanced against a minimization of collisions by the connection with existing objects, including other connections. The method is described in connection with the generation of graphical models of finite state machines (FSMs), but is suitable for use in any system where it is desirable to rapidly produce connected graphs and to maintain a minimum, user-definable level of clarity and legibility of the graph.

Efficient detection and protection of information in cross tabulated tables I: linear invariant test

Abstract: To protect sensitive information in a cross tabulated table, it is a common practice to suppress some of the cells in the table. A linear combination of suppressed cells is called a linear invariant if the combination has a unique feasible value. Intuitively, the information contained in an invariant is not protected even though the values of the suppressed cells are not disclosed. This paper gives a surprisingly efficient algorithm for testing whether a linear combination of suppressed cells is an invariant. In sequential computation, the algorithm runs in optimal linear time. In parallel computation, the algorithm runs in polylogarithmic time using a polynomial number of processors on a parallel random access machine. The algorithm exploits a linear algebraic structure of directed and undirected cycles in a mixed graph induced by a given table. This new structure also plays a crucial role in subsequent papers on other aspects of detecting and protecting sensitive information in a cross tabulated table.

Parallel complexity of the connected subgraph problem

Abstract: This paper shows that the problem of testing whether a graph G contains an induced subgraph of vertex (edge) connectivity at least k is P-complete for any fixed $k \geqslant 3$. Moreover, if $k_{\max } $ is the largest vertex (edge) connectivity of any subgraph of G, it is shown that unless ${\text{P}} = {\text{NC}}$ there is no NC algorithm that approximates $k_{\max } $ within any approximation factor $\frac{1}{2} < c < 1$ (such an algorithm is by definition one that outputs a number in the interval $[ck_{\max } ,k_{\max } ]$). In contrast, it is known that the problem of finding the Tutte (triconnected) components of G (i.e., the maximal subgraphs of G such that for any four vertices in any of them, any two of these vertices can be connected by a path in G that avoids the other two) is in NC. On the positive side, it is shown, by proving extremal graph results, that the maximum k for which there is a k-edge-connected induced subgraph of G can be approximated in NC for any approximation factor strictly ...

A general framework for developing adaptive fault-tolerant routing algorithms

Abstract: It is shown that Cartesian product (CP) graph-based network methods provide a useful framework for the design of reliable parallel computer systems. Given component networks with prespecified connectivity, more complex networks with known connectivity and terminal reliability can be developed. CP networks provide systematic techniques for developing reliable fault-tolerant routing schemes, even for very complex topological structures. The authors establish the theoretical foundations that relate the connectivity of a CP network, the connectivity of the component networks, and the number of faulty components: present an adaptive generic algorithm that can perform successful point-to-point routing in the presence of faults: synthesize, using the theoretical results, this adaptive fault-tolerant algorithm from algorithms written for the component networks: prove the correctness of the algorithm: and show that the algorithm ensures following an optimal path, in the presence of many faults, with high probability. >

Point set domination number of a graph

Abstract: A set D of vertices in a connected graph G is a `psd-set' if for every set S subset-or-equal-to V-D there exists a vertex nu is-an-element-of D such that the subgraph [S or {nu}] induced by S or {nu} is connected. The point-set domination number' gamma(p)(G) of G is the minimum cardinality of a psd-set. Besides some bounds, exact values of gamma(p)(G) are determined when G is a tree, block graph and cactus. A generalization of gamma(p)(G) is also considered.

A note on the last new vertex visited by a random walk

Abstract: A “cover tour” of a connected graph G from a vertex x is a random walk that begins at x, moves at each step with equal probability to any neighbor of its current vertex, and ends when it has hit every vertex of G. The cycle Cn is well known to have the curious property that a cover tour from any vertex is equally likely to end at any other vertex; the complete graph Kn shares this property, trivially, by symmetry. Ronald L. Graham has asked whether there are any other graphs with this property; we show that there are not. © 1993 John Wiley & Sons, Inc.

A Vertex-To-Vertex Pursuit Game Played with Disjoint Sets of Edges

Abstract: The problem is to determine the number of ‘cops’ needed to capture a ‘robber’ where the game is played with perfect information, the different sides moving alternately. The ‘cops’ capture the ‘robber’ if one of them occupies the same vertex as the robber at any time in the game. Normally, both sides can move along any edge present in the graph. We investigate the game where the two sides move along disjoint sets of edges. Two natural situations occur. One, given a graph, the cops move along the edges of the graph and the robber moves along the complementary edges. Two, the adversaries can move along disjoint sets of edges present in a product of graphs.

Graphical Degree Sequence Problems with Connectivity Requirements

Abstract: A sequence of integers D=(d1, d2,..., dn) is k-conneted graphical if there is a k-connected graph with vertices v1, v2,..., vn such that deg(vi)=di for each i=1,2,..., n. The k-connected graphical degree sequence problem is: Given a sequence D of integers, determine whether it is k-connected graphical or not, and, if so, construct a graph with D as its degree sequence. In this paper, we consider the k-connected graphical degree sequence problem and present an O(n log log n) time algorithm.

On the search for the best correlation between graph theoretical invariants and physicochemical properties

Abstract: We have examined the correlation of the Randic connectivity index with the Hosoya topological index, the Wiener number and the molecular identification number in search of the optimal functional relation between those indices and the boiling points of alkanes. We found that some functional relations used empirically in the literature can be understood using the known fact that the Randic connectivity index is the most successful single descriptor of molecular structure.

On topological and geometrical distance matrices

Abstract: Differences between topological and geometrical distance matrices are examined. Some examples of geometrical distances when graphs are embedded in spaces of different dimensions are given. Relations of topological distance matrices to other graph matrices are shown. The topological distance matrices are defined in the Hilbert space and their elements are distances through the graph lattices.