Showing papers on "Null graph published in 1993"

Recontamination does not help to search a graph

Abstract: This paper is concerned with a game on graphs called graph searching. The object of this game is to clear all edges of a contaminated graph. Clearing is achieved by moving searchers, a kind of token, along the edges of the graph according to clearing rules. Certain search strategies cause edges that have been cleared to become contaminated again. Megiddo et al. [9] conjectured that every graph can be searched using a minimum number of searchers without this recontamination occurring, that is, without clearing any edge twice. In this paper, this conjecture is proved. This places the graph-searching problem in NP, completing the proof by Megiddo et al. that the graph-searching problem is NP-complete. Furthermore, by eliminating the need to consider recontamination, this result simplifies the analysis of searcher requirements with respect to other properties of graphs.

The Small Index Property for ω‐Stable (ω‐Categorical Structures and for the Random Graph

Abstract: We give a criterion involving existence of many generic sequences of automorphisms for a countable structure to have the small index property. We use it to show that (i) any a>-stable co-categorical structure, and (ii) the random graph have the small index property. We also show that the automorphism group of such a structure is not the union of a countable chain of proper subgroups.

Transitive closure algorithms based on graph traversal

Abstract: Several graph-based algorithms have been proposed in the literature to compute the transitive closure of a directed graph. We develop two new algorithms (Basic_TC and Gobal_DFTC) and compare the performance of their implementations in a disk-based environment with a well-known graph-based algorithm proposed by Schmitz. Our algorithms use depth-first search to traverse a graph and a technique called marking to avoid processing some of the arcs in the graph. They compute the closure by processing nodes in reverse topological order, building descendent sets by adding the descendent sets of children. While the details of these algorithms differ considerably, one important difference among them is the time at which descendent set additions are performed. Basic_TC, results in superior performance. The first reason is that early additions result in larger descendent set sizes on the average over the duration of the execution, thereby causing more I/O; very often this turns out to more than offset the gains of not having to fetch certain sets again to add them. The second reason is that information collected in the first pass can be used to apply several optimizations in the second pass. To the extent possible, we also adapt these algorithms to perform path computations. Again, our performance comparison confirms the trends seen in reachability queries. Taken in conjunction with another performance study our results indicate that all graph-based algorithms significantly outperform other types of algorithms such as Seminaive and Warren.

An O(n2 algorithm for circular-arc graph recognition

Abstract: In this paper we demonstrate a close relationship between circular-arc graphs and chordal bipartite graphs. We exploit this relationship to achieve an O(n2) time circular-arc graph recognition algorithm. The algorithm employs a new 0 (n2) time method for computing neighborhood containment relations in circular-am graphs and reductions to the recognition problem on circular-arc graphs whose vertices can be partitioned into two cliques. The recognition algorithm also yields an 0 (n2) time isomorphism test for circular-arc graphs.

Basic subgraphs and graph spectra.

Abstract: Every symmetric matrix has an invertible principal submatrix whose rank is equal to that of the whole matrix. We explore some of the implications of this result for graph spectra. U sing the same symbol for a graph and its adjacency matrix, we say that a graph G is A-invertible if G AI is invertible, and the A-rank of G is the rank of G AI. A A-basic subgraph of G is an induced A-invertible subgraph H of the same A-rank as G. Using A-basic subgraphs, we prove that if A 1:. {O, -I} then the set of graphs of a given A-rank is finite; this can be extended to A = 0 and A = -1 by excluding graphs with what we call duplicate and coduplicate vertices respectively. We give an algorithm to construct the graphs of a given A-rank. We show how O-basic sub graphs can be used to calculate ranks of graphs, using as an example graphs obtained by adding two vertices to a complete graph. We examine some properties of maxima/reduced graphs, graphs which occur in characterising the graphs of a given rank, and construct some infinite families of such graphs.

Maintaining the 3-edge-connected components of a graph on-line

Abstract: The problem of maintaining the 3-edge-connected components of a graph undergoing repeated dynamic modifications, such as edge and vertex insertions, is studied. This paper shows how to answer the question of whether or not two vertices belong to the same 3-edge-connected component of a connected graph that is undergoing only edge insertions. Any sequence of q query and updates on an n-vertex graph can be performed in $O((n + q)\alpha (q,n))$ time.

Computing the perspective projection aspect graph of solids of revolution

Abstract: An algorithm for computing the aspect graph for a class of curved-surface objects based on an exact parcellation of 3-D viewpoint space is presented. The object class considered is solids of revolution. A detailed analysis of the visual events for this object class is given, as well as an algorithm for constructing the aspect graph. Numerical search techniques, based on a geometric interpretation of the visual events, have been devised to determine those visual event surfaces that cannot be calculated directly. The worst-case complexity of the number of cells in the parcellation of 3-D viewpoint space, and, hence, the number of nodes in the aspect graph, is O(N/sup 4/), where N is the degree of a polynomial that defines the object shape. A summary of the results for 20 different object descriptions is presented, along with a detailed example for a flower vase. >

Parallel Program Graphs and their Classification

Abstract: We categorize and compare different representations of program dependence graphs, including the Control Flow Graph (CFG) which is a sequential representation lacking data dependences, the Program Dependence Graph (PDG) which is a parallel representation of a sequential program and is comprised of control and data dependences, and more generally, the Parallel Program Graph (PPG) which is a parallel representation of sequential and (inherently) parallel programs and is comprised of parallel control flow edges and synchronization edges. PPGs are classified according to their graphical structure and properties related to deadlock detection and serializability.

AGG - An Implementation of Algebraic Graph Rewriting

Abstract: The Agg-system (Algebraic Graph Grammar System) is a prototype implementation of the algebraic approach to graph transformation [Ehr79]. It has been programmed in EIFFEL and runs on SUN workstations under X Window 11.5. It consists of a flexible graphical editor and a derivation component. The editor allows the graphical manipulation of rules, redices and derivation results. The derivation component performs direct transformation steps for user-selected rules and redices.

Abstract Graph Derivations in the Double Pushout Approach

Abstract: In the algebraic theory of graph grammars, it is common practice to present some notions or results “up to isomorphism”. This allows one to reason about graphs and graph derivations without worrying about representation-dependent details.

On-line Graph Algorithms for Incremental Compilation

Abstract: Compilers usually construct various data structures which often vary only slightly from compilation run to compilation run. This paper describes in a compact and uniform way solutions to several problems arising in order to quickly update these data structures instead of building them from scratch each time. All the considered problems can be reduced to graph problems. Specifically, we give algorithms for the dynamic problems of loop detection, topological order, reachability from the start routine, and transitive closure.

Amalgamated Graph Transformations and Their Use for Specifying AGG - an Algebraic Graph Grammar System

Abstract: The Agg-system is a prototype implementation of the algebraic approach to graph transformation. It consists of a flexible graph editor and a transformation component. The editor allows the graphical representation of production rules, occurrences and transformation results. The transformation component performs direct transformation steps for user-selected production rules and occurrences.

On the nonuniform complexity of the graph isomorphism problem

Abstract: The nonuniform complexity of the graph isomorphism (GI) and graph automorphism (GA) problems is studied, and the implications of different types of polynomial-time reducibilities from these problems to sparse sets are considered. It is shown that if GI (or GA) is bounded truth-table or conjunctively reducible to a sparse set, then it is in P; if it is supposed that it is truth-table reducible without restrictions to a sparse set (or, equivalently, that it belongs to P/poly), then the problem is low for MA, the class of sets with publishable proofs. With respect to nondeterministic reductions, it is shown that if the considered problems are btt strong nondeterministically reducible to a sparse set, then they are in co-NP. Some of these results are proved using graph constructions that show properties of the GI and GA problems. >

Multicyclic components in a random graph process

Abstract: A limit theorem is obtained for the number of components with more than one cycle that are created during the evolution of a random graph process. In particular, it is shown that the probability that the process never contains more than one such component is about 0.87 © 1993 John Wiley & Sons, Inc. © 1993 Wiley Periodicals, Inc.

Efficient decision procedures for graph properties on context-free graph languages

Abstract: Efficient ways of analyzing families of graphs that are generated by a certain type of context-free graph grammars are considered. These graph grammars are called cellular graph grammars. They generate the same graph families as hyperedge replacement systems, but are defined in a way that supports complexity analysis. A characteristic called “finiteness” of graph properties are defined, and combinatorial algorithms are presented for deciding whether a graph language generated by a given cellular graph grammar contains a graph with a given finite graph property. Structural parameters are introduced that bound the complexity of the decision procedure and special cases for which the decision can be made in polynomial time are discussed. Extensions to graph grammars that are not context-free are also given. Our results provide explicit and efficient combinatorial algorithms where, so far, only the existence of algorithms has been shown, or the best known algorithms are highly inefficient.

Graph color minimization using neural networks

Abstract: The problem considered here is that of register allocation which has been shown to map to the non-planar graph coloring problem. Given a graph G, the problem is to color, or label, the vertices such that no two adjacent vertices are the same color. The neural network model presented also minimizes the number of colors used. This additional capability provides for a higher efficiency of resource usage as well as having application to other problems that map to graph coloring. Test results are compared with previous neural network techniques for graph coloring, and it is shown that this neural network model consistently will provide a significant reduction in the number of colors, or registers, required while maintaining a high convergence rate.

Incresing the Vertex-Connectivity in Directed Graphs

Abstract: Given a k-vertex-connected directed graph G, what is the minimum number m, such that G can be made k+1-connected by the addition of m new edges? We prove that if a vertex v has in- and out-degree at least k+1, there exists a splittable pair of edges on v. With the help of this statement, we generalize the basic result of Eswaran and Tarjan, and give lower and upper bounds for m which are equal for k=0 and differ from each other by at most k otherwise. Furthermore, a polynomial approximation algorithm is given for finding an almost optimal augmenting set.