Distance-related Properties of Corona of Certain Graphs

A graph G is called a m−eccentric point graph if each point of G has exactly m ≥ 1 eccentric points. When m = 1, G is called a unique eccentric point (u.e.p) graph. Using the notion of corona of graphs, we show that there exists a m−eccentric point graph for every m ≥ 1. Also, the eccentric graph Ge of a graph G is a graph with the same points as those of G and in which two points u and v are adjacent if and only if either u is an eccentric point of v or v is an eccentric point of u in G. We obtain the structure of the eccentric graph of corona G ◦ H of self-centered or non-self-centered u.e.p graph G with any other graph H and obtain its domination number

Subclasses of harmonic mappings defined by convolution

AbstractTwo new subclasses of harmonic univalent functions defined by convolution are introduced. The subclasses generate a number of known subclasses of harmonic mappings, and thus provide a unified treatment in the study of these subclasses. Sufficient coefficient conditions are obtained that are shown to be also necessary when the analytic parts of the harmonic functions have negative coefficients. Growth estimates and extreme points are also determined

Two-Dimensional Digitized Picture Arrays and Parikh Matrices

Parikh matrix mapping or Parikh matrix of a word has been introduced in the literature to count the scattered subwords in the word. Several properties of a Parikh matrix have been extensively investigated. A picture array is a two-dimensional connected digitized rectangular array consisting of a finite number of pixels with each pixel in a cell having a label from a finite alphabet. Here we extend the notion of Parikh matrix of a word to a picture array and associate with it two kinds of Parikh matrices, called row Parikh matrix and column Parikh matrix. Two picture arrays A and B are defined to be M-equivalent if their row Parikh matrices are the same and their column Parikh matrices are the same. This enables to extend the notion of M-ambiguity to a picture array. In the binary and ternary cases, conditions that ensure M-ambiguity are then obtained

Flat Splicing Array Grammar Systems Generating Picture Arrays

While studying the recombinant behaviour of DNA molecules, Head (1987) introduced a new operation, called splicing on words or strings, which are finite sequences of symbols. There has been intensive research using the concept of splicing on strings in the context of DNA computing, establishing important theoretical results on computational universality. A particular class of splicing, known as flat splicing on strings was recently considered and this operation was extended to provide picture array generating two-dimensional models. Making use of the operation of flat splicing on arrays, we propose here a grammar system, called flat splicing regular array grammar system (FSRAGS), as a new model of picture generation. The components of a FSRAGS generate picture arrays working in parallel using the rules of a two-phase grammar called 2RLG and with two different components of the FSRAGS communicating using the array flat splicing operations on columns and rows of the arrays. We establish some comparison results bringing out the generative power of FSRAGS and also exhibit the power of FSRAGS in generating certain “floor designs”

Algebraic Properties of Parikh Matrices of Words under an Extension of Thue Morphism

The Parikh matrix of a word $w$ over an alphabet $\{a_1, \cdots , a_k \}$ with an ordering $a_1 < a_2 < \cdots a_k,$ gives the number of occurrences of each factor of the word $a_1 \cdots a_k$ as a (scattered) subword of the word $w.$ Two words $u,v$ are said to be $M-$equivalent, if the Parikh matrices of $u$ and $v$ are the same. On the other hand properties of image words under different morphisms have been studied in the context of subwords and Parikh matrices. Here an extension to three letters, introduced by S$\acute{e}\acute{e}$bold (2003), of the well-known Thue morphism on two letters, is considered and properties of Parikh matrices of morphic images of words are investigated. The significance of the contribution is that various classes of binary words are obtained whose images are $M-$equivalent under this extended morphism

A Variant of Extended Two-dimensional Context-free Picture Grammar

Here we introduce a variant of extended two-dimensional context-free picture grammar (E2DCF P G), called (l/u)E2DCF P G which allows rewriting only the leftmost column, or the uppermost row of variables in a picture array. Several theoretical properties of (l/u)E2DCF P G are obtained and an application in generating digitized picture arrays is discussed

Array P Systems and t−Communication

The two areas of grammar systems and P systems, which have provided interesting computational models in the study of formal string language theory have been in the recent past effectively linked in [4] by incorporating into P systems, a communication mode called t−mode of cooperating distributed grammar systems. On the other hand cooperating array grammar systems [5]and array P systems [1] have been developed in the context of two-dimensional picture description. In this paper, motivated by the study of [4], these two systems are studied by linking them through the t−communication mode, thus bringing out the picture description power of these systems

Two-Dimensional Picture Grammar Models.

A new theoretical model of grammatical picture generation called extended 2D context-free picture grammar (E2DCFPG) generating rectangular picture arrays of symbols is introduced. This model which allows variables in the grammar and uses the squeezing mechanism of forming the picture language over terminal symbols, is an extension of the pure 2D context-free picture grammar (P2DCFPG) [13]. The extended picture grammar model E2DCFPG is shown to have more picture generative power than the P2DCFPG and certain other existing 2D models. Certain closure and other properties of this new model are also examine

Pure 2D picture grammars and languages

A new syntactic model, called pure two-dimensional (2D) context-free grammar (P2DCFG), is introduced based on the notion of pure context-free string grammar. The rectangular picture generative power of this 2D grammar model is investigated. Certain closure properties are obtained. An analogue of this 2D grammar model called pure 2D hexagonal context-free grammar (P2DHCFG) is also considered to generate hexagonal picture arrays on triangular grids

Language generating alphabetic flat splicing P systems

An operation on strings, called at splicing was introduced, inspired by a splicing operation on circular strings considered in the study of modelling of the recombinant behaviour of DNA molecules. A simple kind of at splicing, called alphabetic at splicing, allows insertion of a word with a specified start symbol and/or a specified end symbol, between two pre-determined symbols in a given word. In this work, we consider a P system with only alphabetic at splicing rules as the evolution rules and strings of symbols as objects in its regions. We examine the language generative power of the resulting alphabetic at splicing P systems (AFS P systems, for short). In particular, we show that AFS P systems with two membranes are more powerful in generative power than AFS P systems with a single membrane. We also construct AFS P systems with at most three membranes to generate languages that do not belong to certain other language classes and show an application to generation of chain code pictures