Showing papers on "Null graph published in 1984"

Succinct representations of graphs

Abstract: For a fixed graph property Q , the complexity of the problem: Given a graph G , does G have property Q ? is usually investigated as a function of | V |, the number of vertices in G , with the assumption that the input size is polynomial in | V |. In this paper the complexity of these problems is investigated when the input graph is given by a succinct representation. By a succinct representation it is meant that the input size is polylog in | V |. It is shown that graph problems which are approached this way become intractable. Actually, no “nontrivial” problem could be found which can be solved in polynomial time. The main result is characterizing a large class of graph properties for which the respective “succinct problem” is NP-hard. Trying to locate these problems within the P-Time hierarchy shows that the succinct versions of polynomially equivalent problems may not be polynomially equivalent.

On multicommodity flows in planar graphs

Abstract: Okamura and Seymour recently proved two properties of multicommodity flows in undirected planar networks where all the sources and the sinks are on a common face of the underlying graph One is that a feasible solution is guaranteed whenever each cut's capacity is at least as large as the cut's demand The second is that if all demands and capacities are integers then the flow values may be chosen half-integer-valued In this paper we use the first property to construct two computational procedures; one examines the existence of a feasible flow, and the other constructs such a flow if one exists We also show that the construction procedure can be used as an alternative proof to the above properties Finally we show, by presenting counterexamples, that the half-integrality property does not necessarily hold when either the graph cannot be drawn in the plane with all sources and sinks on a common face, or the graph is directed

Pseudo-Boolean functions and stability of graphs

Abstract: In this Daper the concept of conflict graph associated with a pseudo-Boolean function is discussed; one exploits the fact that the problem of finding a stable set with maximum weight in a graph can be reduced to the maximisation of a pseudo-Boolean function and conversely On these Boolean foundations a graph theoretical procedure is developed associating to any graph another one having a strictly smaller stability number Fragmentary computational experience seems to show that this reduction may be applied efficiently in algorithms for obtaining the stability number of a graph

Efficient Implementation Of Graph Algorithms Using Contraction

Abstract: We define a graph problem which we refer to as the component merging problem. Versions of the problem appear as bottlenecks in various graph algorithms. We show how to solve an important special case of the problem.

Partitioning, Spectra and Linear Programming

Abstract: We obtain lower bounds on the value of the objective in a graph partitioning problem in terms of spectral information about the graph. Our approach introduces a class of generalized transportation problems which can be solved in a greedy fashion using the Monge sequence idea.

A Representation of Hypergraphs in the Euclidean Space

Abstract: This paper introduces a graph space that shows concisely the relative weights among combinations of vertices of a given hypergraph. (A hypergraph is a graph in which one edge may connect two or more vertices.) The hypergraph is represented by a collection of points in graph space such that the distance between vertices in graph space reflects the weights of the edges between vertices of the original hypergraph. Vertices of the hypergraph that are connected by edges with large weights are mapped to nearby points in graph space. Thus, graph space reveals properties of the connectivity of vertices in the hypergraph. A natural application of graph space is the placement of modules in computer systems since strongly coupled modules are transformed into nearby points in graph space. The graph of the airlines network in the United States is taken as an example of a hypergraph, and the paper illustrates the corresponding graph space.

Upward topological analysis of large circuits using directed graph representation

Abstract: This paper presents the method of topological analysis of large LLS networks with the use of hierarchical decomposition of the network graph. It is assumed that the network is represented by a directed graph. An algorithm of upward hierarchical analysis of a partitioned graph is presented. The algorithm allows symbolic analysis of large networks with the number of elements kept as symbols practically unlimited. The computational time linearly depends on the network size. A computer program using techniques described is also presented.

Probabilistic methods in graph theory

Abstract: One of the lessons Paul Erdos taught us is that probabilistic counting arguments often yield surprisingly strong existence results in combinatorics. This paper illustrates the paradigm on four examples drawn from Erdos's own work. The examples concern the chromatic number of a graph.

Layouts for the shuffle-exchange graph based on the complex plane diagram

Abstract: The shuffle-exchange graph is one of the best structures known for parallel computation. Among the things, a shuffle-exchange computer can be used to compute discrete Fourier transforms, multiply matrices, evaluate polynomials, perform permutations and sort lists. The algorithms needed for these operations are quite simple and many require no more than logarithmic time and space per processor. In this paper, we analyze the algebraic structure of the shuffle-exchange graph in order to find area-efficient embeddings of the graph in a two-dimensional grid. The results are applicable to the design of Very Large Scale Integration (VLSI) circuit layouts for a shuffle-exchange computer.

A five-color theorem for graphs on surfaces

Abstract: We prove that if a graph embeds on a surface with all edges suitably short, then the vertices of the graph can be five-colored. The motivation is that a graph embedded with short edges is locally a planar graph and hence should not require many more than four colors.

Encoding graphs by derivations and implications for the theory of graph grammars

Abstract: A typical (notion of a sequential) graph grammar G consists of a f i n i t e set of labels z, a set of terminal labels A, (A ~ z) , a f i n i t e set of productions of the form YI ~ Y2' where YI and Y2 are graphs (with labels from z) , and a s ta r t graph (or a f i n i t e set of s ta r t graphs). A der ivat ion step in G is performed as fol lows. Given a graph X and a production Y1 ~ Y2 from G, one locates a subgraph of X isomorphic to YI and "replaces" i t by a subgraph Y1⁄2 isomorphic to Y2" The crucial part of the replacement is to establ ish connections between Y' and the remainder of ×. , 2 The way that the connections are established is specif ied by the so-called embeddingmechanism which may be unique for the whole grammar or i n t r i n s i c to each of the productions. This embedding mechanism is rea l l y "the heart of G". Often also appl icat ion conditions are added to the productions in G roughly speaking, they specify which subgraphs of × that are isomorphic to Y1 may be replaced. The language generated by G is the set of a l l graphs labeled by terminal labels only which can be derived from a s ta r t graph in one or more steps. (See Rosenfeld & Milgram, 19~2; Della Vigna & Ghezzi, 1978; Nagl, 1979; Ehrig, 1979; or Janssens & Rozenberg, 1980, 1982, for examples of d i f fe ren t types of graph grammars and embedding mechanisms.) We give here a somewhat informal presentation of a very simple idea which is well applicable to (almost) every graph grammar concept independently of the embedding mechanism used. Given a graph X in a graph language generated by a graph grammar G, we encode this graph by encoding i t s der ivat ion. In general, such an endoding w i l l be more "cOmplex" than the standard representation of X by i t s nodes, edges, and labels. However, i f the der ivat ion of the graph is "reasonably short" , then th is encoding outperforms the standard representation. This simple observation has a number of implications for normal forms of graph grammars. In par t icu lar , we show that a graph grammar which generates a l l graphs ( la beled by some arb i t ra ry but f ixed set of labels) cannot be essent ia l l y growing.

A directed graph representation based on a statistical hypothesis testing and application to citation and association structures

Abstract: A statistical hypothesis testing method is developed for generating a directed graph from a matrix of nonsymmetric interactions among elements of a system. Two threshold parameters are applied to an interaction matrix in order to generate edges of the graph. Chi-square goodness-of-fit test is considered. The generated graph is plotted on a plane as a figure with hierarchically order vertices. Two applications are considered: one is the structure of literature in computer applications and cybernetics based on citation relationships; the other is the structure of scientific notions of students based on an association test in educational psychology.

Graph factors with given properties

Abstract: We present a sufficient condition for a graph to have a (g,f)-factor which contains p given edges but does not contain other q given edges, where g and f are integer-valued functions defined on the vertices of the graph.

Optimal Parallel Algorithms for Graph Connectivity.

Abstract: : We give a new randomized parallel RAM algorithm for finding a spanning forest of an undirected graph in logarithmic time. These time bounds hold with arbitrary high probability for any input graph (i.e., we do not assume random input; these bounds hold for the worst case input graph). This result assumes a parallel RAM model which allows both concurrent writes and concurrent reads. Furthermore, we show that if the graph is not very sparse (i.e., if the number of edges is at least a logarithmic squared factor more than the number of vertices) than we can achieve a linear processor time product (even for logarithmic time bounds) for finding a spanning tree--which is optimal for the parallel RAM model. Furthermore, we can also achieve a linear processor, time product for even sparser graphs with only slight time increase. Keywords include: graph connectivity, parallel algorithms, optimal algorithms, randomized algorithms.

Module Positioning Algorithms for Rectilinear Macrocell Assemblies

Abstract: A completely hierarchical approach to integrated circuit design begins by partitioning a design problem into subproblems which are based on functional boundaries. It is desirable to produce a final layout which is compact, yet preserves the functional decomposition. Allowing the physical macrocells to have arbitrary rectilinear shapes permits this goal to be achieved but introduces many levels of complexity into the modeling of the assembly. To support macrocells with rectilinear shapes, a directed graph, referred to as an adjacency graph is used to model the positional relationship of the components in the assembly. Algorithms are presented for constructing the adjacency graphs, identifying the cycles present in the adjacency graph, converting the graph to an acyclic graph, and for establishing the component and channel positions based on a critical path analysis. These algorithms are implemented in Pascal on a DECSYSTEM-20.

The conjugacy problem for graph products with central cyclic edge groups

Abstract: A graph product is the fundamental group of a graph of groups. Amongst the simplest examples are HNN extensions and free products with amalgamation. Graph products with cyclic edge groups inherit a solvable conjugacy prob- lem from their vertex groups under certain conditions, the most important of which imposed here is that all the edge group generators in each vertex group are powers of a common central element. Under these conditions the conju- gacy problem is solvable for any two elements not both of zero reduced length in the graph product, and for arbitrary pairs of elements in HNN extensions, tree products and many graph products over finite-leaf roses. The conjugacy problem is not solvable in general for elements of zero reduced length in graph products over graphs with infinitely many circuits. 1. Introduction. A solvable conjugacy problem (S.C.P.) is generally not in- herited by graph products of groups with S.C.P. (see Miller (8)). If attention is re- stricted to graph products with cyclic edge groups, more may be said. It is unlikely that this restriction can be lifted (cf. (6, p. 387; 7, p. 114)). Finite groups, finitely generated free groups, finitely generated nilpotent groups, one-relator groups with torsion or nontrivial centre and certain small cancellation groups all have S.C.P., so there is a wealth of potential vertex groups which may be used in constructing such graph products. In (4) the author shows that a recursively presented graph product with cyclic edge groups over a finite graph inherits S.C.P. from its vertex groups if the sets of cyclic generators in them are "semicritical", thus generalising (5, 7) for HNN extensions and free products with amalgamation. However, semicriticality is a very restrictive condition, which does not hold if all the cyclic generators in a vertex group are powers of a common element. Such cases occur often enough: the celebrated Baumslag-Solitar non-Hopfian groups fall in this category. Here this complementary case is considered. Not surprisingly, a further condition is imposed on the sets of cyclic generators: that they are central in their respective vertex groups. This is suggested by the direct proof that the Baumslag-Solitar groups have S.C.P., and by (2, §3). It is comparatively straightforward to show that the conjugacy problem is solvable for any two elements in the graph product of which at least one has nonzero reduced length (Theorem 3.1). For elements of zero reduced length the problem is much more difficult, reducing to the question of whether a specific recursively enumerable (r.e.) set is recursive (Theorem 3.3).

Line Graph of Gamma-Acyclic Database Schems and its Recognition Algorithm

Abstract: In this paper the properties of the line graph of y-aeyclic hyperg-raphs are described. Based on the properties, an efficient algorithm is given for determining whether a hypergraph is γ-acyclic. The algorithm runs in 0(n(n+e)) time for a hypergraph with its line graph having n vertices and e edges.

Evaluation of Graph Representations with Active Nodes.

Abstract: In recent years, the reduction scheme is widely recognized as a good implementing method of functional languages. In most parallel reduction methods, it is required to inspect and update a program graph in a common store. However, this results in memory conflictions and makes a trouble in constructing a correct reducer. This paper presents a method for reducing a graph by each active node. In our method, the graph is divided and distributed to each local memory of a processing unit, and is reduced by a process assigned to each node. Since reduction is performed in each node, parallel reduction is realized without a common store.