Graph-theoretic Results Research Articles

Vortex dynamics, combinatorics and statistics

This report represents an overview of the interconnections between the dynamics of large vortex systems, combinatorics, n-body problems and statistical mechanics. The combinatorial perturbation method for the 2D vortex problem is discussed; the essential combinatorial symplectic transformations to Jacobi-type variables which are based on a binary tree algorithm, is introduced and extended to the 3D vortex problem. Combinatorial and graph-theoretic results which are motivated by the computational needs of the vortex problem, are mentioned. They include new results on sign-nonsingular patterns and noneven digraphs. A simplified singular limit of the 3D Hamiltonian for vortex dynamics is derived and its basic properties discussed. The 2- and 3-body problems in this simple model is studied.

Computer-aided generation of high-temperature series expansions for mixed-spin Ising model ferrimagnets

We describe a computer algorithm, justified by graph-theoretic results, which aids the calculation of high-temperature series coefficients for Ising models. This algorithm has been efficiently implemented on a AMT DAP, and preliminary results for mixed-spin Ising model spinel ferrimagnets are presented here.

Spanners in graphs of bounded degree

AbstractGiven a graph G = (V, E), a subgraph S = (V, Es) is a t‐spanner of G if for every edge xy ϵ E the distance between x and y in S is at most t. Spanners have applications in communication networks, distributed systems, parallel computation, and many other areas. This paper is concerned with the complexity of finding a minimum size t‐spanner in a graph with bounded degree. A linear time algorithm is presented for finding a minimum‐size 2‐spanner in any graph whose maximum degree is at most four. The algorithm is based on a graph theoretical result concerning edge partition of a graph into a “triangle‐free component” and “triangular‐components.” On the other hand, it is shown that to determine whether a graph with maximum degree at most nine contains a t‐spanner with at most K edges (K is given) is NP‐complete for any fixed t ⩾ 2. © 1994 by John Wiley & Sons, Inc.

Path Coverings of Graphs and Height Characteristics of Matrices

Using graph theoretic techniques, it is shown that the height characteristic of a triangular matrix A majorizes the dual sequence of the sequence of differences of maximal cardinalities of singular k-paths in the graph G( A) of A and that in the generic case the height characteristic is equal to that dual sequence. The results on matrices are also used to prove a graph theoretic result on the duality of the sequence of differences of minimal k th norms of path coverings for a (0-1)-weighted acyclic graph G and the sequence of differences of maximal cardinalitics of k-paths in G. This result generalizes known results on unweighted graphs.

The relation between the Jordan structure of a matrix and its graph

It is shown that the height characteristics of a matrix A strongly majorizes the dual sequence of the sequence of differences of maximal cardinalities of nonclosable k-paths in G( A), and that in the generic case the height characteristics is equal to that dual sequence. Simultaneously, it is shown that the sequence of differences of minimal kth nonclosable norms of path coverings for a directed graph G is the dual of the sequence of differences of maximal cardinalities of nonclosable k-paths in G. These results generalize both matrix theoretical and graph theoretical known results for the triangular case.

Digital Jordan curves — a graph-theoretical approach to a topological theorem

We give a proof of a graph-theoretical Jordan curve theorem which generalizes both the topological results of Khalimsky et al. as well as some of the graph-theoretical results of Rosenfeld and Klette and Voss.

Aircraft and maintenance scheduling support, mathematical insights and a proposed interactive system

In this paper we first review the current practice of operative aircraft and maintenance scheduling at the Hungarian Airlines. The ideas to be included in the proposed operative scheduling support system mean new contributions from both the algorithmic and human-computer interaction points of view. The algorithm is based on new graph theoretical results which were motivated by the necessity of combining a given flight schedule with the strict maintenance requirements of the aircraft. Aircraft rotation corresponds in mathematical terms to the coloring of an interval graph with colors representing the tail numbers in such a way that the vertices corresponding to maintenance checks are colored in advance. The human-computer interface is built on the Microsoft Windows graphics environment, which allows a simultaneous, visual and active contact with all necessary information and methods.

A decision procedure revisited: notes on direct logic, linear logic and its implementation

This paper studies decidable fragments of predicate calculus. We will focus on the structure of direct predicate calculus as defined in Ketonen and Weyhrauch (1984) in the light of the recent work of Girard (Girard 1987, 1988) on linear logic. Several graph-theoretic results are used to prove correspondences between systems of natural deduction, direct predicate logic, and linear logic. In addition, the implementation of a decision procedure for direct predicate logic is sketched.

Structured system models Part 2. Directed graphs and boolean matrices

Parameterized system models and their augmented controllability indices are studied. Analytically parameterized system models arc shown to have generic index lists, i.e. lists which arc actual lists for all parameter values except those in a proper analytic variety. For structured system models, as a subset of analytically parameterized system models, there exists a known graph-theoretic method for determining the Hermite indices. A new method for examining both Hermite and Kronecker indices is introduced. The method is based on a necessary graph-theoretic condition for the examined columns of the controllability matrix to be linearly independent. Additional sufficient conditions are given to complete the method. The given graph-theoretic results are also expressed using boolean matrices associated with graphs.

Colouring the discretization graphs arising in the multigrid method

We formulate aims and conditions of graph colourings suitable for the (triangulation) graphs arising in the finite element-based multigrid method, review graph-theoretical results, describe linear-time 6-, 5- and 4-colour algorithms and give results of numerical experiments.

Edge disjoint spanning trees in random graphs

We show that almost every G . contains m edge disjoint spanning m—out trees. Introduction In this note we consider the maximum number of edge disjoint spanning trees contained in the random graph G _ . Let a graph G = (V,E) have property sL if it contains spanning trees T1tTo,...,T, which are pair-wise edge disjoint. We consider the random graph G = G . This has vertex set V = m m-out n {l,2,...,n}. Each v € V independently chooses a set out(v) of distinct vertices as neighbours, where each m-subset of V -{v} is equally likely to be chosen. This produces a random m out-regular diagraph D which has m been selected uniformly from ( ) distinct possibilities. G is obtained by ignoring orientation but without coalescing edges. (See [1], [2], [3] for properties of this model.) Probability statements refer to the probability space of D and graph theoretic statements refer to G . m Theorem 1 Let m > 2 be a fixed constant. Then lim Pr(G € st ) = 1. I n^> m m [This is clearly best possible.] The major graph theoretic result underpinning our proof is as follows. University Libraries "arnegie Mellon University rtfh.PA f5213-3890 Theorem 2 (Nash-Williams [5], Tutte [6]) A graph G = (V,E) has property sL if and only if for every partition SlfSo,...,S of V, 2 < t < |v|, there at least k(t-l) edges of G joining vertices in different subsets of the partition. D (The necessity of the condition is obvious. The "meat" is in the sufficiency.) Proof of main result For S C V let -Y(S) = |{vw € E(D ): v € S, w CS} I. "~ n m

Recognizing circle graphs in polynomial time

The main result of this paper is an 0 ([ V ] x [ E ]) time algorithm for deciding whether a given graph is a circle graph, that is, the intersection graph of a set of chords on a circle. The algorithm utilizes two new graph-theoretic results, regarding necessary induced subgraphs of graphs having neither articulation points nor similar pairs of vertices. Furthermore, as a substep of the algorithm, it is shown how to find in 0 ([ V ] x [ E ]) time a decomposition of a graph into prime graphs, thereby improving on a result of Cunningham.

On some convexities

The aim of the present paper is to investigate some convexities. First we introduce and investigate the "pseudoconvex" spaces and give a minimax theorem applying the ideas of [8]. The notion of convex spaces was introduced by Komiya [1] (see below for details). In [2] the authors proved a Nikaidd--Isoda-type theorem for convex spaces. An interesting convexity notion, not giving a convex space in the sense of [1] is obtained in [5]. The paper [4] contains investigations with respect to this convexity structure, analogous to that of in [2]. Now we give a common generalization of the convexities of [i] and [5], the so-called pseudoconvex space. It turns out that the compact pseudoconvex spaces have the fixed point property and a Nikaido-Isoda-type theorem holds. First we prove these assertions and after we prove an inequality between the Helly and Caratheodory number of pseudoconvex spaces. In connection with these investigations we mention the work of M. Horv~th [3] where the Helly, Caratheodory and Radon numbers of a special convexity structure are calculated and whose proof is based on graph theoretical results. At last we continue the investigation of V. Komornik [10] and give a related minimax theorem in interval space.

A concept for parameter independent evaluation of decentralized stabilizability

A major problem due to the use of decentralized feedback for large scale or complex technical plants is the question of stabilizability, i.e. the existence of local controllers stabilizing the overall system. The paper describes a concept for investigating whether a plant, which consists of state-coupled SISO-subsystems, can be stabilized by local dynamic output feedback of fixed dynamic order. The investigation is carried out in a parameter independent way based on graph-theoretic results such that stabilizability is guaranteed irrespective of the numerical values of the non-vanishing system parameters. The approach permits the analysis of systems with more general structures than methods known up to now.

Graph theoretic reliability analysis for the Boolean n cube networks

Two graph-theoretic results concerning Boolean n-cube network reliability are presented. First, a simple formula for the number of spanning trees of the Boolean n-cube network is derived. As a result, the reliability function for large failure rate can be readily computed. Second, the Boolean n-cube network is proved to have the super-line-connectivity property. Thus the number of line-disconnecting sets (a set of lines the removal of which results in a disconnected or trivial graph) or order lambda for the Boolean n-cube network is equal to 2/sup n/.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Parameter Independent Evaluation of Decentralized Stabilizability

A major problem due to the use of decentralized feedback for large scale or complex technical plants is the question of stabilizability, i. e. the existence of local controllers stabilizing the overall system. The paper describes a concept for investigating whether a plant, which consists of state-coupled SISO-subsystems, can be stabilized by local dynamic output feedbacks of fixed dynamic order. The investigation is carried out in a parameter independent way based on graph theoretic results such that stabilizability is guaranteed irrespective of the numerical values of the non-vanishing system parameters. The approach permits the analysis of systems with more general structures than known methods. The significance of the proposed algorithm is demonstrated by means of a hydroelectric power system.

Fundamentals of planar ordered sets

A finite ordered set is planar when it can be represented in the plane with the point a lower than b in the plane whenever a is less than b in the ordered set, and with a straight-line segment from a to b whenever a is covered by b. We prove it is equivalent to allow instead an arc from a to b as long as the arc does not go below the level of a or above the level of b (whenever a is covered by b). Our result is analogous to the well-known graph-theoretic result of K. Wagner and I. Fáry.

Row-ordering schemes for sparse givens transformations. II. implicit graph model

A new graph model is presented to study the row annihilation and row ordering problems in the QR decomposition of sparse matrices using Givens rotations. The graph-theoretic results obtained can be used to derive good row orderings for certain column orderings, such as width-1 and width-2 nested dissection orderings. This model is different from the bipartite-graph model introduced in [6]. We refer to the new model as implicit because the rows are not represented explicitly by nodes, in contrast to the bipartite-graph model, where the rows are represented by nodes in a bipartite graph.

The properties of connected graphs and some corrections to the list of Uhlenbeck and Ford

Some graph-theoretical results for connected cluster integrals are used to correct six entries in the table of Uhlenbeck and Ford.(1)

Nowhere zero flow and circuit covering in regular matroids

Let s( k) be the smallest s such that if there exists a k-nowhere zero flow in a regular matroid M, then M can be covered by circuits, the total length of which is at most s| M|. A recursive formula for the evaluation of s( k) is given: s(kt) ≤ (s(k)(kt − t) + s(t)(kt − k)) (kt − 1) . By means of this formula s( k) is found for k = 2, 3, 4, 6, 7, 8. “Natural” proofs for graph theoretical results are obtained.