17739753
Finite Languages for the Representation of Finite Graphs
We introduce a new way of specifying graphs: through languages, i.e., sets of strings. The strings of a given (finite, prefix-free) language represent the vertices of the graph; whether or not there is an edge between the vertices represented by two strings is determined by the pair of symbols at the first position in these strings where they differ. With this new, “positional” or lexicographic, method, classical and well-understood ways of specifying languages can now be used to specify graphs, in a compact way; thus, (small) finite automata can be used to specify (large) graphs. Since (prefix-free) languages can be viewed as trees, our method generalizes the hierarchical specification of particular types of graphs such as cographs and VSP graphs. Our main results demonstrate an intrinsic relationship between the fundamental operations of language concatenation and graph substitution.
39431424
Incremental construction of 2-structures
Abstract A labeled 2-structure g is an edge-labeled directed graph with a reversibility condition on the labels. Each labeled 2-structure g can be represented as a tree, the prime tree of g. We characterize a labeled 2-structure h in terms of the prime tree of a labeled 2-structure g which results from h by removing one element from h. This leads to an O(n3) algorithm for the construction of the prime tree.
34951842
Primitivity is Hereditary for 2-Structures
Abstract It is proved that each primitive 2-structure on n elements ( n ⩾3) contains a primitive substructure on either ( n −1) or ( n −2) elements.
37770747
Transitivity of local complementation and switching on graphs
Abstract The operations complementation C , local complementation λ x , and switching σ x for the vertices x of a finite undirected graph are considered. The operation λ x complements the subgraph induced by the neighbourhood of x in the given graph, and the switching σ x changes the neighbourhood of x to its complement vertex set. It is proved that the compositions δ x = λ x C (for vertices x ∈ D ) generate a transitive group on the graphs with vertex set D , that is, for any two graphs g and h on D , there exists a composition α of operations δ x such that h = α ( g ). It is also shown that the compositions τ x = λ x σ x (for x ∈ D ) generate a transitive group on the graphs.
12269815
Angular 2-Structures
The theory of 2-structures forms a convenient framework for investigating graphs. In this paper we investigate the class of angular 2-structures which results by requiring that the primitivity is forbidden on the lowest possible level, i.e. it is required that no substructure on three elements is primitive. The paper presents the basic theory of angular 2-structures and, in particular, the theory of primitive angular 2-structures. We demonstrate that the notion of an angular 2-structure is a well-chosen generalization of the notions of a symmetric graph and a partial order.
40876494
Theory of 2-Structures
The process according to the present invention relates to the production of ferrovanadium alloys. The process comprises melting a charge containing iron and vanadium with a subsequent reduction of iron and vanadium to give a metal melt. In accordance with the present invention, iron and vanadium are reduced substantially completely. The resulting metal melt is poured into a converter and then purged, through the converter bottom, by means of oxygen blowing at a supply rate of oxygen and for a period of time sufficient to produce a slag containing at least 35% by weight of vanadium pentoxide. Thereafter, the metal melt is discharged from the converter, while vanadium and iron are reduced from the resulting slag by the metallothermal method.
17444444
The Bounded Degree Problem for NLC Grammars is Decidable
Abstract The degree of a graph H is the maximum among the degrees of its nodes. A set of graphs L is of bounded degree if there exists a positive integer n such that the degree of each graph in L does not exceed n . We demonstrate that it is decidable whether or not the (graph) language of an arbitrary node label controlled (NLC) grammar is of bounded degree. Moreover, it is shown that, given an arbitrary NLC grammar G generating the language L(G) of bounded degree, one can effectively compute the maximum integer which appears as the degree of a graph in L(G) .
205092173
CTS Systems and Petri Nets
The general theory of coordinated table selective substitution systems (cts systems for short) (see Rozenberg (1985)), provides a unifying framework for a considerable number of grammar and automaton models considered in the literature. This paper is mainly devoted to the investigation of a- natural subclass of cts systems (which uses the 'context-free grammar selector' for its memory access) and it turns out that this subclass closely corresponds to the Petri net model of concurrent processes.
43370814
Boundary NLC Graph Grammars-Basic Definitions, Normal Forms, and Complexity
Node label controlled (NLC) grammars are graph grammars (operating on node labeled undirected graphs) which rewrite single nodes only and establish connections between the embedded graph and the neighbors of the rewritten node on the basis of the labels of the involved nodes only. They define (possibly infinite) languages of undirected node labeled graphs (or, if we just omit the labels, languages of unlabeled graphs). Here we consider a restriction of NLC grammars, so-called boundary NLC (BNLC) grammars , distinguished by the property that whenever in a graph already generated two nodes may be rewritten, then these nodes are not adjacent. The graph languages generated by this type of grammars are called BNLC languages . Although we show that this restriction leads to a smaller class of languages, still enough generative power remains to define interesting graph languages. For example, trees, complete bipartite graphs, maximal outerplanar graphs, k -trees, graphs of bandwidth ⩽ k , graphs of cyclic bandwidth ⩽ k , graphs of binary tree bandwidth ⩽ k , graphs of cutwidth ⩽ k (always for a fixed positive integer k ) turn out all to be BNLC languages. We prove a number of normal forms for BNLC grammars and then we indicate their usefulness by various applications. In particular, we show that for connected graphs of bounded degree the membership problem for BNLC languages is solvable in deterministic polynomial time.
11853375
Edge-Label Controlled Graph Grammars
Abstract We introduce a graph-grammar model based on edge-replacement, where both the rewriting and the embedding mechanisms are controlled by edge labels. The general power of this model is established—it turns out to have the complete power of recursive enumerability (in a sense to be made precise in the paper). In order to understand where this power originates, we identify three basic features of the embedding mechanism and examine how restrictions on these features affect the generative power. In particular, by imposing restrictions on all three features simultaneously, we obtain a graph-grammar model that was previously introduced by Kreowski and Habel.
206851807
Behaviour of Elementary Net Systems