Definition Of Graph Research Articles

On the termination of clause graph resolution

This paper introducesextended clause graph resolution, a variant of Kowalski's clause graph resolution that is terminating at the full first-order level. This terminating variant is obtained by extending the definitions of clause graph and clause graph resolution to include more information about the interdependencies between links and clauses in the graph, by restricting purity slightly and by employing an exhaustive search of eligible links.

Structure graphs and structure graph grammars

In this paper we give the formal definition of a structure graph and formalize the concepts of structure parameters and graph theoretical properties of structure graphs. Structure graphs are a formal descriptive method that can model three-dimensional objects. We introduce various extensions of structure graph grammars, a class of graph grammars that generates structure graphs, and compare the classes of languages of the structure graph grammar extensions.

The cordiality of one-point union of n copies of a graph

In this paper we give an equivalent definition of a cordial graph. The definition implies a previous result of Cahit (1986); it also enables us to find infinite families of noncordial graphs, derive some bound on the number of edges in a cordial graph and establish a necessary and sufficient condition for a one-point union of two n-cliques. Let G be a rooted graph. We denote by G ( n) the graph obtained from n copies of G by identifying their roots. A sufficient condition for G ( n) to be cordial is related to the solution of a system involving one equation and two inequalities with their coefficients depending on some binary labellings of G. According to the solvability of the system, we are able to establish a number of necessary and sufficient conditions for the cordiality of G ( n) for certain classes of G, such as cycles, complete graphs, wheels, fans and flags.

The Cartesian product of a k-extendable and an l-extendable graph is ( k + l + 1)-extendable

Abstract : Let us start with the definition of a kappa-extendable graph G. Suppose kappa is an integer such that 1 < or = kappa < or = (/V(G)/-2)/2. A graph G is kappa-extendable if G is connected, has a perfect matching (a 1- factor) and any matching in G consisting of kappa edges can be extended to (i.e. , is a subset of) a perfect matching. The extendability number of G, extG, is the maximum kappa such that G is kappa-extendable. A natural problem is to determine the extendability number of a graph G.

Reaction graphs and a construction of reaction networks

A general definition of reaction graphs is presented. For a pair of isomeric molecular graphs \(\mathbb{G}_1 \) and \(\mathbb{G}_2 \), related by a chemical transformation \(\mathbb{G}_1 {\text{ }} \Rightarrow {\text{ }}\mathbb{G}_2 \), the reaction graph \(\mathbb{G}_R^{{\text{(}}\omega {\text{)}}} \) is determined using a maximal common subgraph defined for vertex mapping \(\omega :V(\mathbb{G}_1 ) \to V(\mathbb{G}_2 )\). A binary operation ‘ ⊕ ’ defined for graphs constructed over the same vertex set enables us to decompose the reaction graph \(\mathbb{G}_R^{(\omega )} \) into the sum of prototype reaction graphs. A decomposition of an overall reaction graph can be advantageously used for the construction of a reaction network. An oriented path in this network beginning at \(\mathbb{G}_1 \) and ending at \(\mathbb{G}_2 \) corresponds to a breakdown of the transformation \(\mathbb{G}_1 {\text{ }} \Rightarrow {\text{ }}\mathbb{G}_2 \) into a sequence of intermediates.

On the geometry of Zara graphs

We show that, up to trivial examples, the definition of Zara graph (or a graph satisfying hypothesis (H) in F. Zara [ European J. Combin. 5 (1984) , 255–290]) can be weakened to the following axiom: (Z) every maximal clique is regular in the sense of A. Neumaier [ in “Finite Geometries and Designs” (P. J. Cameron, J. W. P. Hirschfeld, and D. R. Hugues, Eds.), pp. 244–259, Cambridge Univ. Press, Cambridge, U. K., 1981] (i.e., given a maximal clique, every vertex outside it is adjacent to the same number of vertices in it).

The mathematical foundations of bond graphs—I. Algebraic theory

Elementary mathematical concepts from linear algebra are used to develop an independent theory for non-directed bond graphs. The definition of a bond graph is given and its structure as a combinatorial object is studied. This is accomplished by constructing a vector space, called the bond space of the bond graph. Topological information is encoded in the bond graph by defining a subspace of the bond space, the s-space. The internal bond information is eliminated by studying subspaces of the s-space, the cycle and internal space. A precise meaning is given to the association between bond graphs and linear graphs by comparing their respective cycle spaces. A technique, called the cut and paste method, is described which produces a bond graph associated with a given graph via its planar representation. Several problems associated with the inverse procedure are discussed.

A proof by graphs that PSL(2, 7) ≅ PSL(3, 2)

We give three definitions of the Coxeter graph. By the second one we see that PSL(2, 7) is contained in the automorphism group of that graph as a subgroup of index 2, and by the third one that the same holds for PSL(3, 2).

On infinite perfect graphs and randomized stopping points on the plane

We show that perfection of polytopes defined by infinite zero-one matrices can be characterized as in the finite case through an appropriate definition of infinite perfect graphs. We examine the relationship between certain such polytopes and a convex set of random measures on the discrete plane, termed randomized stopping points, that appear in the context of the optimal stopping problem for two-parameter processes. Using probabilistic techniques, we show that under a conditional qualitative independence hypothesis on the underlying filtration, which we express as a commutation property of a “conditional supremum” operator, the only extremal elements of this set of random measures are ordinary stopping points.

Theory and computational applications of Fibonacci graphs

The concept of Fibonacci graphs introduced and developed by this author is critically reviewed. The concept has been shown to provide an easypencil-and-paper method of calculating characteristic, matching, counting, sextet, rook, color and king polynomials of graphs of quite large size with limited connectivities. For example, the coefficients of the matching polynomial of 18-annuleno—18-annulene can be obtainedby hand using the definition of Fibonacci graphs. They are (in absolute magnitudes): 1, 35, 557, 5337, 34 361, 157 081, 525 296, 1304 426, 2 416 571, 3 327 037, 3 362 528, 2 440 842, 1 229 614, 407 814, 81936, 8652, 361, 3. The theory of Fibonacci graphs is reviewed in an easy and detailed language. The theory leads to modulation of the polynomial of a graph with the polynomial of a path.

Link inheritance in abstract clause graphs

Clause graphs, as they were defined in the 1970s, are graphs representing first order formulas in conjunctive normal form: the nodes are labelled with literals and the edges (links) connect complementary unifiable literals, i.e. they provide an explicit representation of the resolution possibilities. This report describes a generalization of this concept, called abstract clause graphs. The nodes of abstract clause graphs are still labelled with literals, the links however connect literals that are ‘unifiable’ relative to a given relation between literals. This relation is not explicitely defined, only certain abstract properties are required. For instance the existence of a special purpose unification algorithm is assumed, which computes substitutions, the application of which makes the relation hold for two literals. When instances of already existing literals are added to the graph (e.g. due to resolution or factoring), the links to the new literals are derived from the links of their ancestors. An inheritance mechanism for such links is presented which operates only on the attached substitutions and does not have to unify the literals. The definition of abstract clause graphs and the theory about link inheritance is general enough to provide a framework so that as new ideas are proposed for graph based theorem provers, the methodology for both implementing links and proving their properties will be readily available.

FIFO Nets and Modelling of Manufacturing Processes in FMS

Timed FIFO nets are proposed for simulation of manufacturing processes in MS. Subclass of FIFO nets - normal FIFO nets-is defined for which boundedness, liveness and reachability like problems are solvable. The notion of controlled FIFO nets is introduced to obtain a unified description of material and informational processes. The definition of the operational graph of manufacturing system, description of some problems and their solutions are given in terms of FIFO nets for illustration of the potential of proposed means.

Computing Geometric Properties of Images Represented by Linear Quadtrees

The region quadtree is a hierarchical data structure that finds use in applications such as image processing, computer graphics, pattern recognition, robotics, and cartography. In order to save space, a number of pointerless quadtree representations (termed linear quadtrees) have been proposed. One representation maintains the nodes in a list ordered according to a preorder traversal of the quadtree. Using such an image representation and a graph definition of a quadtree, a general algorithm to compute geometric image properties such as the perimeter, the Euler number, and the connected components of an image is developed and analyzed. The algorithm differs from the conventional approaches to images represented by quadtrees in that it does not make use of neighbor finding methods that require the location of a nearest common ancestor. Instead, it makes use of a staircase-like data structure to represent the blocks that have been already processed. The worst-case execution time of the algorithm, when used to compute the perimeter, is proportional to the number of leaf nodes in the quadtree, which is optimal. For an image of size 2n × 2n, the perimeter algorithm requires only four arrays of 2n positions each for working storage. This makes it well suited to processing linear quadtrees residing in secondary storage. Implementation experience has confirmed its superiority to existing approaches to computing geometric properties for images represented by quadtrees.

Series - parallel graphs and depth-first search trees

Series-parallel graphs play a significant role in the analysis and synthesis of electrical networks, communication networks, and switching circuits. On the other hand, due to their very tractable structure, a number of algorithmic problems which are NP-complete for arbitrary graphs can be efficiently solved for the special case of series-parallel graphs, see some recent results in [7]. In another recent work, Shinoda et al. [6] have shown that series-parallel graphs can be completely characterized by a property of their spanning trees. They proved, that every spanning tree of a series-parallel graph G is a depth-first search (or DFS) tree of a 2-isomorphic copy of G . The proof in [6] is, however, existential. The purpose of this note is to provide a constructive proof of this property. We present a procedure which for a given series-parallel graph G and its spanning tree T , produces a 2-isomorphic copy G' of G such that the edges of T generate in G' a DFS tree of G' . Our considerations are entirely based on the classical constructive definition of series-parallel graphs and the Duffin's characterization, unlike the work [6], where several other characterizations of these graphs are utilized. We refer the reader to Chen [1] and Harary [4] for graph-theoretic terms not defined here.

Program package GRADAP: Graph definition and analysis package

Graph analysis is applicable in every situation in which (points) have relations with (lines toward) other objects and in which the structure of these relations is to be studied. Some examples are: relations between persons (family, friends, commercial relations), relations between functions (e.g., functions fulfilled by the same person), and functions between political parties and states, where a common border can be considered as the relation defining the graph. The aim of graph analysis is the detection of patterns in such networks, the interdependence of patterns formed by different relations, the implications of patterns for the behavior of objects, and the impact of the qualities of the objects on the patterns. Graph analysis, therefore, consists of methods that, on given graphs, try to find: subgraphs (subsets of points in the graph) that meet certain requirements of relations (like strong relations), partial graphs (all points, with a subset of lines) in selecting relations, relations between such subgraphs, detection of trees in the graph (e.g., the shortest spanning tree), and detection of paths (e.g., the shortest path). Of course, the analysis becomes more complicated when: a directed graph is used (lines run in only one direction), the graph is signed (each directed line has a positive or negative sign), the points and/or lines valued (e.g., the distance between two points as a value for the connecting line), or more than one type of graph is used at the same time (a multigraph). Also, more than one relation between points may be considered in the same analysis. Secondary analysis may consist of the analysis of properties of points within detected clusters, the comparison of clusters, and so on. A simple example of a graph is the sociogram in Figure 1. In Figure 1, A to I represent children in a class. An arrow from A toward B represents A choosing B as a playmate. Other relations, defined for the same children, could be previously studied together, are members of the same club, and so on. Additional information includes age, sex, school results, and so on. An example of additional information is a preference order

Historical Software Issue 6: Graph Definition and Analysis Package (GRADAP)

Historical Software Issue 6: Graph Definition and Analysis Package (GRADAP)

Characterization for Series-Parallel Channel Graphs

Channel graphs are often used to describe the connecting paths between a pair of terminals in a link system. Series-parallel (SP) channel graphs and Takagi graphs are special classes of channel graphs heavily studied in the literature. In this paper, we give a noninductive definition of SP channel graphs which we prove to be equivalent to the usual inductive definition. We then use this new definition to characterize SP Takagi graphs and regular SP channel graphs.

Trunk division methods in the channel routing style

AbstractIn the layout design of building block LSI's wires are placed in the region between a pair of columns of cells. Usually, lines parallel to columns of cells (trunk) and lines perpendicular to them (branch) are placed in different layers and they are connected by throughholes. In the channel routing method we note first that the ordering of trunks is restricted if branches of different nets should not intersect each other. Then we represent this restriction with a digraph called a constraint graph. Using this graph we determine the order of placements of trunks. When there are directed circuits in a constraint graph we must destroy them by partitioning trunks.In this paper we modify the definition of constraint graph so that transformations on a constraint graph due to a partition of trunks become easy. To be more specific, information on terminals also will be incorporated in a constraint graph thereby limiting the extent of a transformation due to a partition of trunks. In addition, the trunk division method proposed earlier is redefined considering multiple divisions; conditions for each partition method to be effective in removing conflicting constraints are examined.

Random Graphs: Structural-Contextual Dichotomy

A formal definition of random graphs is introduced which is applicable to graphical pattern recognition problems. The definition is used to formulate rigorously the structural-contextual dichotomy of random graphs. The probability of outcome graphs is expressed as the product of two terms, one due to the statistical variability of structure among the outcome graphs and the other due to their contextual variability. Expressions are obtained to estimate the various probability, typicality, and entropy measures. The members in an ensemble of signed digraphs are interpreted as outcome graphs of a random graph. The synthesized random graph is used to quantify the structural, contextual, and overall typicality of the outcome graphs with respect to the random graph.

Topological Ligand Design. I. Definition and Derivation of Simple Ligand Graph

Abstract A simple ligand graph and a pattern graph are defined topologically in order to design the patterns of the chelating ligands. This ligand graph, which comprises m donor atoms and n carbon chains capable of forming medium-sized chelate rings, is an irregular graph containing neither multiple lines nor self-loops. All the ligand graphs with m≤8 can be systematically obtained from fifty pattern graphs. These are derived from fifteen basic pattern graphs, which are cubic general graphs containing some multiple lines and self-loops. A pattern code and a set of series indices are tabulated for each ligand graph with m≤8. For the ligand graphs with m≤6, all the geometric isomers of their octahedral chelates are also illustrated.