Showing papers on "Null graph published in 1992"

Conditions for unique graph realizations

Abstract: The graph realization problem is that of computing the relative locations of a set of vertices placed in Euclidean space, relying only upon some set of inter-vertex distance measurements. This paper is concerned with the closely related problem of determining whether or not a graph has a unique realization. Both these problems are NP-hard, but the proofs rely upon special combinations of edge lengths. If one assumes the vertex locations are unrelated, then the uniqueness question can be approached from a purely graph theoretic angle that ignores edge lengths. This paper identifies three necessary graph theoretic conditions for a graph to have a unique realization in any dimension. Efficient sequential and NC algorithms are presented for each condition, although these algorithms have very different flavors in different dimensions.

Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families

Abstract: This paper describes a predicate calculus in which graph problems can be expressed. Any problem possessing such an expression can be solved in linear time on any recursively constructed graph, once its decomposition tree is known. Moreover, the linear-time algorithm can be generatedautomatically from the expression, because all our theorems are proved constructively. The calculus is founded upon a short list of particularly primitive predicates, which in turn are combined by fundamental logical operations. This framework is rich enough to include the vast majority of known linear-time solvable problems. We have obtained these results independently of similar results by Courcelle [11], [12], through utilization of the framework of Bernet al. [6]. We believe our formalism is more practical for programmers who would implement the automatic generation machinery, and more readily understood by many theorists.

Random Graph Processes with Degree Restrictions

Abstract: Suppose that a process begins with n isolated vertices, to which edges are added randomly one by one so that the maximum degree of the induced graph is always bounded above by d We prove that if n → ∞ with d fixed, then with probability tending to 1, the final result of this process is a graph with ⌊nd / 2⌋ edges

The even cycle problem for directed graphs

Abstract: If each arc in a strongly connected directed graph of minimum in- degree and outdegree at least 3 is assigned a weight 0 or 1, then the resulting weighted directed graph has a directed cycle of even total weight. This proves a conjecture made by L. Lovisz in 1975 and has applications to colour-critical hypergraphs, sign-nonsingular matrices, and permanents of matrices. MATHEMATICAL INSTITUTE, TECHNICAL UNIVERSITY OF DENMARK, DK-2800 LYNGBY,

A simple algorithm to construct a consistent extension of a partially oriented graph

Abstract: A Partially directed acyclic graph, (pdag), is a graph which contains both directed and undirected edges, with no directed cycle in its directed subgraph. An oriented extension of a pdag G is a fully directed acyclic graph (dag) on the same underlying set of edges, with the same orientation on the directed subgraph ofG and the same set of vee-structures. A vee-structure is formed by two edges, directed toward a common head, while their tails are nonadjacent. A simple polynomial-time algorithm is presented, to solve the following problem: Given a pdag, does it admit an oriented extension? The problem was stated by Verma and Pearl, while studying the existence of causal explanation to a given set of observed independencies.

A tourist guide through Treewidth

Abstract: A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that find tree-decompositions, algorithms that use tree-decompositions to solve hard problems efficiently, graph minor theory, and some applications. The paper contains an extensive bibliography.

Finding k disjoint paths in a directed planar graph

Abstract: It is shown that, for each fixed $k$, the problem of finding $k$ pairwise vertex-disjoint directed paths between given pairs of terminals in a directed planar graph is solvable in polynomial time.

A generalized state assignment theory for transformations on signal transition graphs

Abstract: A constraint satisfaction framework is proposed that can guarantee necessary and sufficient conditions for a state graph assignment to result in a transformed state graph that is race-free. Performing transformations at the state graph level has the advantage that the requirements imposed on the initial signal transition graph (STG) are very weak. Unlike previous methods, the initial STG need not be a live, safe, free choice net. The only requirement is that the corresponding initial state graph should be finite and connected, and have a consistent state assignment. Hence, a very broad range of STGs can be synthesized. The transformation achievable using the proposed framework correspond to very complex transformations on STGs. Even transformations that convert a free choice net into a correct non-free choice net, and a 1-safe net into a correct 2-safe net are feasible. Addition of transitions that do not follow the Petri net firing rule is also possible. >

The g -neighborhood graph

Abstract: This paper presents a novel two-parameter geometric graph, the γ-neighborhood graph This graph unifies a number of geometric graphs such as the convex hull, the Delaunay triangulation, and in 2D also the Gabriel graph and the circle-based β-skeleton, into a continuous spectrum of geometric graphs that ranges from the void to the complete graph The two parameters provide for a great flexibility in the analysis of a set of sites For specific ranges of the parameters, the corresponding graph can be efficiently constructed

A Polynomial-Time Graph Algorithm to Decide Liveness of Some Basic Classes of Bounded Petri Nets

Abstract: This paper is related to structural analysis of Petri nets where liveness and boundedness issues are addressed through the analysis of the combinatorial properties of the underlying graph. We first recall a number of basic results about liveness and boundedness involving combinatorial substructures (deadlocks and traps). It is then shown that testing whether a bounded Extended Free Choice net or a Non Self-Controlling net is structurally live can be reduced to the search for a strongly connected deadlock which is not a trap. This problem, in turn, is shown to be solvable in polynomial time through a purely combinatorial algorithm making combined use of Tarjan's strong connectivity algorithm and Minoux's LTUR algorithm for solving Horn satisfiability problems. Once structural liveness has been proved, testing liveness for a given initial marking is already known to be polynomially solvable.

An apprentice that discovers hypertext links

Abstract: The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without violating planarity. We show how to support each of the above operations in O(n2/3) time, where n is the number of vertices in the graph. The bound for tests and deletions is worst-case, while the bound for insertions is amortized. This is the first algorithm for this problem with sub-linear running time. The same data structure has further applications in maintaining the biconnected and triconnected components of a dynamic planar graph.

A self-stabilizing distributed algorithm to construct bfs spanning trees of a symmetric graph

Abstract: We propose a simple and efficient self-stabilizing distributed algorithm to construct the breadth first search (BFS) spanning tree of an arbitrary connected symmetric graph. We develop a completely new direct approach of graph theoretical reasoning to prove the correctness of our algorithm. The approach seems to have potential to have applications in proving correctness of other self-stabilizing algorithms for graph theoretical problems.

Diagram editors=graphs+attributes+graph grammars

Abstract: This paper reports on the latest developments in ongoing work which started in 1981 and is aimed at a general method which would help to reduce considerably the time necessary to develop a syntax-directed editor for any given diagram technique. In joint projects between the University of Erlangen-Nurnberg and software companies it has been shown that the ideas and the implemented tools can also be used for the design of CAD-systems. Several editors for diagram techniques in the field of software engineering have been implemented (e.g. SDL and SADT). In addition, 3-D-modelling packages for interior design and furnishing or lighting systems have been developed. The main idea behind the approach is to represent diagrams by (formal) graphs whose nodes are enriched with attributes. Then, any manipulation of a diagram (typically the insertion of an arrow, a box, text, coloring etc.) can be expressed in terms of the manipulation of its underlying attributed representation graph. The formal description of the manipulation is done by programmed attributed graph grammars . The main advantage of using graph grammars is the unified approach for the design of the data structures and the representation of the algorithms as graphs and graph productions, respectively. The results proved that graph grammars are a software-engineering method of their own.

Global bifurcation sets and stable projections of nonsingular algebraic surfaces

Abstract: The view graph of a surface is a planar graph whose nodes are the stable views (projections) of the surface and whose edges represent transitional views of codimension one. The space of all directions of orthogonal projection can be identified with the projective plane. The set of “bad” projection directions, associated with the degenerate views of positive codimension, forms a graph in the projective plane (the view bifurcation set). This graph is dual to the view graph and divides the projective plane into a certain number of connected regions whose representatives are the nodes of the view graph. We assume that the projected surface is nonsingular and parameterized by polynomials of degree d. We present an estimate for the number of nodes in the view graph in terms of d and describe symbolic algorithms for computing the bifurcation set and the view graph of a surface from a parametrization.

m -median and m -center problems with mutual communication: solvable special cases

Abstract: In this paper, we consider the network version of the m-median problem with mutual communication (MMMC). We reformulate this problem as a graph theoretic node selection problem defined on a special graph. We give a polynomial time algorithm to solve the node selection problem when the flow graph (graph that denotes the interaction between pairs of new facilities in MMMC) has a special structure. We also show that with some modification in the algorithm for MMMC, the m-center problem with mutual communication can also be solved when the flow graph has a special structure.

On Hajnal's triangle-free game

Abstract: We investigate the triangle-free game proposed by Andras Hajnal. Starting with the empty graph onn points, two players alternatingly pick edges. The loser is the player who is forced to select an edge which completes a triangle. We determine the winner in a version of the game with the additional rule that the chosen edges must always give a connected subgraph ofK n . Some other versions are also investigated.

Shorter Paths to Graph Algorithms

Abstract: We illustrate the use of formal languages and relations in compact formal derivations of some graph algorithms.

Graph Algorithms = Iteration + Data Structures? The Structure of Graph Algorithms and a Corresponding Style of Programming

Abstract: We encourage a specific view of graph algorithms, which can be paraphrased by “iterate over the graph elements in a specific order and perform computations in each step”. Data structures will be used to define the processing order, and we will introduce recursive mapping/array definitions as a vehicle for specifying the desired computations. Being concerned with the problem of implementing graph algorithms, we outline the extension of a functional programming language by graph types and operations. In particular, we explicate an exploration operator that essentially embodies the proposed view of algorithms. Fortunately, the resulting specifications of algorithms, in addition to being compact and declarative, are expected to have an almost as efficient implementation as their imperative counterparts.

The 3-edge-components and a structural description of all 3-edge-cuts in a graph

Abstract: Let G = (V, E) be an undirected graph, ¦V¦ = n. We denote Vl the partition of V into maximal vertex subsets indivisible by k(-edge)-cuts, k < l, of the whole G. The factor-graph of G corresponding to V3, is known to give a clear representation of V2, V3 and of the system of cuts of G with 1 and 2 edges. Here a (graph invariant) structural description of V4 and of the system of 3-cuts in an arbitrary graph G is suggested. It is based on a new concept of the 3-edge-connected components of a graph (with vertex sets from V3). The 3-cuts of G are classified so that the classes are naturally 1∶1 correspondent to the 3-cuts of the 3-edge-connected components. A class can be reconstructed in a simple way from the component cut, using the relation of the component to the system of 2-cuts of G. For 3-cuts and V4 of a 3-edge-connected graph we follow [DKL76]. The space complexity of the description suggested is O(n) (though the total number of 3-cuts may be a cubic function of n).

Applying graph grammars for task-oriented user interface development

Abstract: As a graph with attributes assigned to its nodes is very well suited for the specification of semi-structured application domains, attributed graphs and adequate graph rewriting systems are also useful tools for the specification and representation of user interfaces based on end user tasks. A rewriting system for attributed graphs, which is based on the controlled rewriting of graphs using only six elementary types of graph productions, namely the addition, the deletion and the renaming of a node or an edge, is shown to be a suitable unique model for the specification and the simulation of various kinds of user interfaces. In particular, it bridges the gap between conceptual and implementation-oriented development techniques. >

Algorithms for exploring an unknown graph

Abstract: We consider the problem of exploring an unknown strongly connected directed graph We use the exploration model introduced by Deng and Papadimitriou [DP90] An explorer follows the edges of an unknown graph until she has seen all the edges and vertices of the graph The explorer does not know how many vertices and edges the graph has, or how the vertices are connected At each vertex the explorer can see how many edges are leaving the vertex, but she does not know where they lead to She chooses one such edge and explores it by traversing it Deng and Papadimitriou [DP90] have shown that the graph exploration problem for graphs that are very similar to Eulerian graphs can be solved efficiently They introduce the notion of deficiency for such graphs to measure the "distance" from being Eulerian and give algorithms that solve the exploration problem for deficiency-one and bounded deficiency graphs We review and discuss the problem of exploring an unknown Eulerian graph We carefully describe and analyze an algorithm for deficiency-one graphs that combines the two algorithms that Deng and Papadimitriou [DP90] give for this problem We also briefly discuss the problem of exploring a graph of general deficiency

Connectivity Properties of Matroids

Abstract: The bases-exchange graph of a matroid is the graph whose vertices are the bases of the matroid, and two bases are connected by an edge if and only if one can be obtained from the other by the exchange of a single pair of elements. In this paper we prove that a matroid is "connected" if and only if the "restricted bases-exchange graph" (the bases-exchange graph restricted to exchanges involving only one specific element e) is connected. This provides an alternative definition of matroid connectivity. Moreover, it shows that the connected components of the restricted bases-exchange graph satisfy a "ratios-condition", namely, that the ratio of the number of bases containing e to the number of bases not containing e is the same for each connected component of the restricted bases-exchange graph. We further show that if a more general ratios-condition is also true, namely, that any fraction a of the bases containing e is adjacent to at least a fraction a of the bases not containing e (where a is any real number between 0 and 1), then the bases-exchange graph has the following expansion property: "For any bipartition of its vertices, the number of edges incident to both partition classes is at least as large as the size of the smaller parti".