Abstract: The maximum weight/cardinality stable set problem is to nd a maximum weight/cardinality stable set of a given graph. It is well known that these problems for general graphs belong to the class of NP-hard. However, for several classes of graphs, e.g., for perfect graphs and claw-free graphs and so on, these problem can be solved in polynomial time. For instance, Minty (1980), Sbihi (1980) and Lov asz and Plummer (1986) have proposed polynomial time algorithms nding a maximum cardinality stable set of a claw-free graph. Moreover, it has been believed that Minty's algorithm is the unique algorithm nding a maximum weight stable set of a claw-free graph up to date. Here we show that Minty's algorithm for the weighted version fails for some special cases, and give modi cations to overcome it.

Abstract: Graph grammars may be used as natural and powerful syntax-definition formalisms for visual programming languages. Yet most graph-grammar parsing algorithms presented so far are either unable to recognize interesting visual languages or tend to be inefficient (with exponential time complexity) when applied to graphs with a large number of nodes and edges. This paper presents a context-sensitive graph grammar called reserved graph grammar, which can explicitly and completely describe the syntax of a wide range of diagrams using labeled graphs. The parsing algorithm of a reserved graph grammar uses a marking mechanism to avoid ambiguity in parsing and has polynomial time complexity in most cases. The paper defines a constraint condition under which a graph defined in a reserved graph grammar can be parsed in polynomial time. An algorithm for checking the condition is also provided.

Abstract: We propose a new style of writing graph algorithms in functional languages which is based on an alternative view of graphs as inductively defined data types. We show how this graph model can be implemented efficiently, and then we demonstrate how graph algorithms can be succinctly given by recursive function definitions based on the inductive graph view. We also regard this as a contribution to the teaching of algorithms and data structures in functional languages since we can use the functional-style graph algorithms instead of the imperative algorithms that are dominant today.

Abstract: We identify a set of programming constructs ensuring that a programming language based on graph transformation is computationally complete. These constructs are (1) nondeterministic application of a set of graph transformation rules, (2) sequential composition and (3) iteration. This language is minimal in that omitting either sequential composition or iteration results in a computationally incomplete language. By computational completeness we refer to the ability to compute every computable partial function on labelled graphs. Our completeness proof is based on graph transformation programs which encode arbitrary graphs as strings, simulate Turing machines on these strings, and decode the resulting strings back into graphs.

Abstract: This thesis studies the reconstruction and generation of oriented matroids. Oriented ma¬ troids are a combinatorialabstraction of discrete geometric objects such as point configurations or hyperplane arrangements. Both problems, reconstruction and generation, addressfundamental questions of representing and constructing (classes of) oriented ma¬ troids. The representations which are discussedin this thesis are based on graphs that are definedby the oriented matroids, namely tope graphs and cocircuit graphs. The first part ofthis thesisstudies properties ofthese graphs and the questionas to what extent oriented matroids are determined by these graphs. In the second part, these graph representations are used for the design of generationmethods which produce complete lists of oriented matroids ofgiven number of elements and given rank. Thesegenerationmethods are used in the third part for the construction of a catalog of oriented matroids and of complete listings of the combinatorialtypesof point configurations and hyperplane arrangements. The reconstruction problem is the problem of whether an oriented matroid can be reconstructedfrom somerepresentation of it, which is here the tope graph and the cocir¬ cuit graph. It is known that tope graphs determine oriented matroids up to isomorphism. However,there is no simple graph theoretical characterization of tope graphs of oriented matroids. We strengthen the known properties of tope graphs and prove that for every dement / the topes that are not bounded by / induce a connected subgraph in the tope graph. This propertyis later used for the design of generationmethods that are based on topegraphs. On the contrary to the tope graph case, it is known that cocircuit graphs do not determine isomorphismclasses of oriented matroids. However,if every vertex is labeled by its supporting hyperplane,oriented matroids can be reconstructed up to reorientation.We present a simple algorithmwhich gives a constructive proof for this result. Furthermore, we extend the known results and showthatthe isomorphismclass of auniform oriented matroid is determined by its cocircuitgraph. In addition, we present polynomial algorithmswhich provide a constructive proofto this result, and it is shown that the conectness of the input of the algorithmscan be verifiedin polynomial time. The generationproblem asks for methods for listing all oriented matroids of given cardinality of the ground set and given rank. The known generationmethods have been designed primarily for uniform oriented matroids in rank 3 or 4. Our methods are based on tope graph and cocircuit graph representations and generate all isomorphismclasses of oriented matroids, including non-uniformones in arbitrary rank. The generationap¬ proach incrementallyextends oriented matroids by adding Single elements.These Single

Abstract: The classical algebraic approach to graph transformation is a mathematical theory based on categorical techniques with several interesting applications in computer science. In this paper, a new semantics of graph transformation systems (in the algebraic, double-pushout (DPO) approach) is proposed in order to make them suitable for the specification of concurrent and reactive systems. Classically, a graph transformation system comes with a fixed behavioral interpretation. Firstly, all transformation steps are intended to be completely specified by the rules of the system, that is, there is an implicit frame condition: it is assumed that there is a complete control about the evolution of the system. Hence, the interaction between the system and its (possibly unknown) environment, which is essential in a reactive system, cannot be modeled explicitly. Secondly, each sequence of transformation steps represents a legal computation of the system, and this makes it difficult to model systems with control. The first issue is addressed by providing graph transformation rules with a loose semantics, allowing for unspecified effects which are interpreted as activities of the environment. This is formalized by the notion of double-pullback transitions, which replace (and generalize) the well-known double-pushout diagrams by allowing for spontaneous changes in the context of a rule application. Two characterizations of double-pullback transitions are provided: the first one describes them in terms of extended direct DPO derivations, and the second one as incomplete views of parallel or amalgamated derivations. The issue of constraining the behavior of a system to transformation sequences satisfying certain properties is addressed instead by introducing a general notion of logic of behavioral constraints, which includes instances like start graphs, application and consistency conditions, and temporal logic constraints. The loose semantics of a system with restricted behavior is defined as a category of coalgebras over a suitable functor. Such category has a final object which includes all finite and infinite transition sequences satisfying the constraints.

Abstract: Conceptual graphs are very useful for representing structured knowledge. However, existing formulations of fuzzy conceptual graphs are not suitable for matching images of natural scenes. This paper presents a new variation of fuzzy conceptual graphs that is more suited to image matching. This variant differentiates between a model graph that describes a known scene and an image graph which describes an input image. A new measurement is defined to measure how well a model graph matches an image graph. A fuzzy graph matching algorithm is developed based on error-tolerant subgraph isomorphism. Test results show that the matching algorithm gives very good results for matching images to predefined scene models.

Abstract: Property testing is a relaxation of decision problems in which it is required to distinguish YES-instances (i.e., objects having a predetermined property) from instances that are far from any YES-instance. We present three theorems regarding testing graph properties in the adjacency matrix representation. More specifically, these theorems relate to the project of characterizing graph properties according to the complexity of testing them (in the adjacency matrix representation). The first theorem is that there exist monotone graph properties in /spl Nscr//spl Pscr/ for which testing is very hard (i.e., requires one to examine a constant fraction of the entries in the matrix). The second theorem is that every graph property that can be tested making a number of queries that is independent of the size of the graph, can be so tested by uniformly selecting a set of vertices and accepting iff the induced subgraph has some fixed graph property (which is not necessarily the same as the one being tested). The third theorem refers to the framework of graph partition problems, and is a characterization of the subclass of properties that can be tested using a one-sided error tester, making a number of queries that is independent of the size of the graph.

Abstract: Graph separation is a well-known tool to make (hard) graph problems accessible for a divide and conquer approach. We show how to use graph separator theorems in order to develop fixed parameter algorithms for many well-known NP-hard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a fixed parameter divide and conquer algorithm of running time c√k ċ nO(1) for a constant c.

Abstract: We prove a lower bound of Ω(n4/3 log1/3 n) on the randomized decision tree complexity of any nontrivial monotone n-vertex bipartite graph property, thereby improving the previous bound of Ω(n4/3) due to Hajnal [H91] Our proof works by improving a probabilistic argument in that paper, which also improves a graph packing lemma proved there By a result of Groger [G92] our complexity lower bound carries over from bipartite to general monotone n-vertex graph properties Graph packing being a well-studied subject in its own right, our improved packing lemma and the probabilistic technique used to prove it, may be of independent interest

Abstract: Role assignments, introduced by Everett and Borgatti [2], who called them role colorings, formalize the idea, arising In the theory of social networks, that individuals of the same social role will relate in the same way to the individuals playing counterpart roles. If G is a graph, a k-role assignment is a function r mapping the vertex set onto the set of integers {1,2,…, k} so that if r(x) = r(y) then the sets of roles assigned to the neighbors of x and y are the same. We ask how hard it Is to determine If a graph has a 2-role assignment and show that recognizing if a graph has a 2-role assignment is NP-complete.

Abstract: Let P be a property of graphs. An ∈-test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than ∈(n2) edges to make it satisfy P. The property P is called testable, if for every ∈ there exists an ∈-test for P whose total number of queries is independent of the size of the input graph. Goldreich, Goldwasser and Ron [7] showed that certain graph properties, like k-colorability, admit an ∈-test. In [2] a first step towards a logical characterization of the testable graph properties was made by proving that all first order properties of type “∃∀” are testable while there exist first order graph properties of type “∀∃” which are not testable. For proving the positive part, it was shown that all properties describable by a very general type of coloring problem are testable.While this result is tight from the standpoint of first order expressions, further steps towards the characterization of the testable graph properties can be taken by considering the coloring problem instead. It is proven here that other classes of graph properties, describable by various generalizations of the coloring notion used in [2], are testable, showing that this approach can broaden the understanding of the nature of the testable graph properties. The proof combines some generalizations of the methods used in [2] with additional methods.

Abstract: We define graph products of families of pairs of groups and study the question when two such graph products are commensurable. As an application we prove linearity of certain graph products.

Abstract: A note v of a graph G is called fixed if every automorphism of G sends v onto itself. A graph or digraph or other graphical structure is then called fixed if every node is fixed, i.e., its automorphism group is the identity. We present several methods for fixing a graph (destroying its automorphisms). These may not work for all graphs. The methods include orienting some of the edges, coloring some of the nodes with one or more colors and the same for the edges, labeling nodes or edges, and adding or deleting nodes or edges. These considerations lead to a multitude of new invariants and open questions. If a graph already has the identity group, then it is fixed. If not, then fixing a graph G means altering it in some way to obtain a fixed graphical structure. The first published method of fixing a graph is apparently due to Tutte (13). He applied his procedure to planar graphs only. He selected an arbitrary edge of the graph, oriented it in one of the two possible ways, and then drew at the center of the selected edge a small arrow perpendicular to the edge, oriented in one of the two possible directions. This served to fix the planar graph. The purpose of the small arrow was to specify the exterior region of the graph. Tutte's objective was to count triangulations of the plane without having to take their symmetries into consideration. Our purpose is to present an exposition of several methods of fixing a graph: 1. Some graphs can be fixed by orienting a subset of its edges. This is not true of all

Abstract: The main aim of this paper is to characterize the Green relations in the graph product of monoids. Necessary and sufficient conditions for an element in a graph product of monoids to be idempotent, regular or completely regular, are established. These characterizations immediately lead to decidability results. A new proof for the word problem is also presented.

Abstract: We prove that among finite graph algebras and among finite flat graph algebras, dualizability, full dualizability, strong dualizability and entropicity are all equivalent. Any finite (flat) graph algebra which is not dualizable must be inherently non-\( \kappa \)-dualizable for every infinite cardinal \( \kappa \). A new, general method for proving strong duality is presented.

Abstract: In a 3-dimensional orthogonal drawing of a graph, vertices are mapped to grid points on an integer lattice and edges are routed along integer grid lines. In this paper, we present a layout scheme that draws any graph with n vertices of maximum degree 6, using at most 6 bends per edge and in a volume of O(n2). The advantage of our strategy over other drawing methods is that our method is fully dynamic, allowing both insertion and deletion of vertices and edges, while maintaining the volume and bend bounds. The drawing can be obtained in O(n) time and insertions/deletions can be performed in O(1) time. Multiple edges and self loops are permitted. A more elaborate construction that uses only 5 bends per edge, and a simpler, more balanced layout that requires at most 7 bends per edge are also described.

Abstract: A k-tree of a connected graph is a spanning tree with maximum degree at most k. We obtain a sufficient condition for a graph to have a k-tree, as a generalization of the condition of E. Flandrin, H. A. Jung and H. Li [3] for traceability. We also extend early results of Y. Caro, I. Krasikov and Y. Roditty [2] and Min Aung and Aung Kyaw [4] for the maximal order of a tree with bounded maximum degree in a graph.

Abstract: In this paper an efficient method is developed for nodal and element ordering of structures and finite element models. The present method is based on concepts from algebraic graph theory and comprises of an efficient algorithm for calculating the Fiedler vector of the Laplacian matrix of a graph. The problem of finding the second eigenvalue of the Laplacian matrix is transformed into evaluating the maximal eigenvalue of the complementary Laplacian matrix. An iterative method is then employed to form the eigenvector needed for renumbering the vertices of a graph. An appropriate transformation, maps the vertex ordering of graphs into nodal and element ordering of the finite element models. In order to increase the efficiency of the algebraic graph theoretical method, a multi-level scheme is adopted in which the graph model corresponding to a finite element mesh is coarsened in various levels to reduce the size of the problem. Then an efficient algebraic method is applied and with an uncoarsening process, the final ordering of the graph and hence that of the corresponding finite element model is obtained.

Abstract: We study the domain of two constructive geometric constraint solving techniques. Both deal with constraints represented by a geometric constraint graph. The first technique analyses the graph bottom-up, from the edges to the whole graph. The second technique analyses the graph top-down, from the whole graph to the individual edges. We describe these techniques using abstract reduction systems which simplifies the study of their properties. We present an abstract description of the domain of each technique. Finally, we show that both techniques have the same domain, that is, they solve the same kind of problems defined by geometric constraints.

Abstract: The Laplacian is another important matrix associated with a graph, and the Laplacian spectrum is the spectrum of this matrix. We will consider the relationship between structural properties of a graph and the Laplacian spectrum, in a similar fashion to the spectral graph theory of previous chapters. We will meet Kirchhoff’s expression for the number of spanning trees of a graph as the determinant of the matrix we get by deleting a row and column from the Laplacian. This is one of the oldest results in algebraic graph theory. We will also see how the Laplacian can be used in a number of ways to provide interesting geometric representations of a graph. This is related to work on the Colin de Verdiere number of a graph, which is one of the most important recent developments in graph theory.

Abstract: We consider the problem of maintaining connected components in a set of moving objects using the kinetic data structure (KDS) framework. We assume that the motion of each object can be specified by a low-degree algebraic trajectory; this trajectory, however, can be modified in an on-line fashion. While the objects move continuously, their connectivity changes at discrete times. A straightforward dynamic graph approach for maintaining connectivity of n objects has three shortcomings: the graph can have O(n2) edges, the update bounds are amortized, and the algorithm is very complicated. Our first result shows that the connectivity for a set of n moving hypercubes can be maintained using a very simple, easy to determine graph with O(n) edges. But this graph still requires a general-purpose dynamic graph scheme for connectivity maintenance. Our main result is a simplified connectivity data structure for moving rectangles in the plane. For this special but important case, we are able to overcome all three shortcomings mentioned above: our graph has O(n) edges; our data structure supports updates in O(log2n) worst-case time; and the algorithm and data structures are quite a bit simpler than those based on a general dynamic graph scheme.

Abstract: This paper presents a new circuit theoretical concept based on the principal partition theorem for distributed network management focusing on loops of an information network. To realize a simple network management with the minimum number of local agents, namely the topological degrees of freedom of a graph, a reduced loop agent graph generated by contracting the minimal principal minor is proposed. To investigate the optimal distribution of the loop agents, a theory of tie-set graph is proposed. Considering the total processing load of loop agents, a complexity of a tie-set graph is introduced to obtain the simplest tie-set graph with the minimum complexity. As for the simplest tie-set graph search, an experimental result shows that the computational time depends heavily on the nullity of the original graph. Therefore, a tie-set graph with the smallest nullity is essential for network management. Copyright © 2001 John Wiley & Sons, Ltd.