Paired and Total Domination on the Queen\u27s Graph.

The Queen’s domination problem has a long and rich history. The problem can be simply stated as: What is the minimum number of queens that can be placed on a chessboard so that all squares are attacked or occupied by a queen? The problem has been expanded to include not only the standard 8x8 board, but any rectangular m×n sized board. In this thesis, we consider both paired and total domination versions of this renowned problem

Queen\u27s domination using border squares and (\u3ci\u3eA\u3c/i\u3e,\u3ci\u3eB\u3c/i\u3e)-restricted domination

In this paper we introduce a variant on the long studied, highly entertaining, and very difficult problem of determining the domination number of the queen\u27s chessboard graph, that is, determining how few queens are needed to protect all of the squares of a k by k chessboard. Motivated by the problem of minimum redundance domination, we consider the problem of determining how few queens restricted to squares on the border can be used to protect the entire chessboard. We give exact values of border-queens required for the k by k chessboard when 1≤k≤13. For the general case, we present a lower bound of k(2-9/2k-√(8k2-49k+49)/2k) and an upper bound of k-2. For k=3t+1 we improve the upper bound to 2t+1 if 3t+1 is odd and 2t if 3t+1 is even. We generalize this problem to (A,B)-restricted parameters for vertex subsets A and B of V(G) where, for example, one must use only vertices in A to dominate all of B. Defining upper and lower parameters for independence, domination, and irredundance, we present a generalization of the domination chain of inequalities relating these parameters

The queen's domination problem

The queens graph Qn has the squares of then x n chessboard as its vertices; two squares are adjacent if they are in the same row, column or diagonal. A set D of squares of Qn is a dominating set for Qn if every square of Qn is either in D or adjacent to a square in D. If no two squares of a set I are adjacent then I is an independent set. Let 'J'(Qn) denote the minimum size of a dominating set of Qn and let i(Qn) denote the minimum size of an independent dominating set of Qn. The main purpose of this thesis is to determine new values for'!'( Qn). We begin by discussing the most important known lower bounds for 'J'(Qn) in Chapter 2. In Chapter 3 we state the hitherto known values of 'J'(Qn) and explain how they were determined. We briefly explain how to obtain all non-isomorphic minimum dominating sets for Q8 (listed in Appendix A). It is often useful to study these small dominating sets to look for patterns and possible generalisations. In Chapter 4 we determine new values for')' ( Q69 ) , ')' ( Q77 ), ')' ( Q30 ) and i (Q45 ) by considering asymmetric and symmetric dominating sets for the case n = 4k + 1 and in Chapter 5 we search for dominating sets for the case n = 4k + 3, thus determining the values of 'I' ( Q19) and 'I' (Q31 ). In Chapter 6 we prove the upper bound')' (Qn) :s; 1 8 5n + 0 (1), which is better than known bounds in the literature and in Chapter 7 we consider dominating sets on hexagonal boards. Finally, in Chapter 8 we determine the irredundance number for the hexagonal boards H5 and H7, as well as for Q5 and Q6Mathematical SciencesD.Phil. (Applied Mathematics

FROM IRREDUNDANCE TO ANNIHILATION: A BRIEF OVERVIEW OF SOME DOMINATION PARAMETERS OF GRAPHS

Durante los últimos treinta años, el concepto de dominación en grafos ha levantado un interés impresionante. Una bibliografía reciente sobre el tópico contiene más de 1200 referencias y el número de definiciones nuevas está creciendo continuamente. En vez de intentar dar un catálogo de todas ellas, examinamos las nociones más clásicas e importantes (tales como dominación independiente, dominación irredundante, k-cubrimientos, conjuntos k-dominantes, conjuntos Vecindad Perfecta, ...) y algunos de los resultados más significativos.   PALABRAS CLAVES: Teoría de grafos, Dominación.   ABSTRACT During the last thirty years, the concept of domination in graphs has generated an impressive interest. A recent bibliography on the subject contains more than 1200 references and the number of new definitions is continually increasing. Rather than trying to give a catalogue of all of them, we survey the most classical and important notions (as independent domination, irredundant domination, k-coverings, k-dominating sets, Perfect Neighborhood sets, ...) and some of the most significant results.   KEY WORDS: Graph theory, Domination

Flow field prediction and analysis, project fire final report

Flow field, and convective and radiative heating predictions for NASA Fire project reentry package trajectorie

Accounting Historians Journal, 1986, Vol. 13, no. 1 [whole issue]

Spring issu

Handel and the English Chapel Royal during the reigns of Queen Anne and King George I

Handel's association with the English Chapel Royal began soon after he settled in London, and continued intermittently for the remainder of his life. ' The thesis traces this association until'the death of King George I in 1727. Handel's Chapel Royal music is examined with dual reference to the Chapel's musical traditions and the development of the composer's style. Political events influenced the extent of Handel's involvement with the Chapel at different periods: the relationship between the composers, the Chapel Royal and the British Court is described within the historical contexts drawing on many unpublished sources. After an introductory survey of the repertoire of Handel's Chapel Royal music, the musical history of the Chapel during the period preceding Handel's arrival in London is described, paying special attention to performances of instrumentally-accompanied services and anthems, and the ceremonial Thanksgiving Services of Queen Anne's reign. Handel's experience of church music In Italy and Germany previous to his arrival in England is described, and the contemporary English anthems of William Croft are examined by way of an introduction to Handel's first English Verse Anthem. Two chapters are devoted, to Handel's Chapel Royal compositions from the period 1710-1714, including his music celebrating the Peace of Utrecht. Chapters 6 and 7 investigate the circumstances surrounding Handel's disassociation from the Chapel in 1715-1722 and his subsequent re-establishment there during the last years of George I's reign. The significance, of Handel's appointment as Composer to the Chapel is considered, and a chronology for his Chapel Royal music from the 1720p suggested. The music is examined in detail, and described with reference to related antecedent Chandos Anthems., The later history of Handel's Chapel Royal music is surveyed briefly in the final chapter. Factual background material is collected in an extended supplement and fifteen appendices

The life and works of M Loane

The broad aim of this thesis is to examine the life and works of M.Loane, for although her six social studies are often referred to by historians, no detailed or critical study of her background or her corpus has previously been undertaken. This thesis has been divided into a number of chapters. The introduction will set out the precise details of the study which follows, and provide a rationale for the structure adopted. The next chapter will provide an overview of contemporary social studies of poverty, so that the work of M.Loane can been considered contextually. The biographical study of Miss Loane which follows forms a central part of this thesis: not only will it provide details of her career as a nurse and Queen's [i.e. district] nurse, experience which was essential to the content of all her writings, but, importantly, it will enable previously undisclosed information regarding the clandestine literary partnership which existed between two of the Misses Loane to be revealed. Loane's methodological approach will then be considered, and her reputation as a serious social commentator considered within the framework of recent academic criticism of her methods. Subsequent chapters will concentrate on a thematic examination of The Queen's Poor: this social commentary has been selected for a number of reasons. It was the first to be published and was certainly compiled before Loane retired as a Queen's nurse. Beyond this, it can be considered as representative of all the subsequent social studies, all the others being versions of the same structure, and all utilising the same methodology. The conclusion to this thesis will draw together all the strands of this study and emphasise the breakthroughs which have been made: in particular the prolific nature of this writer and her contribution to our understanding of working-class poverty at the turn of the twentieth century will be emphasised. The appendices will reinforce the content of the main thesis. The first of these will present a working bibliography ofLoane's published work, and will reveal the scale of the corpus produced between 1897 and 1911. The second will provide a working list of contemporary reviews of Loane's nursing publications and social studies. This will provide a contemporary view of the response to her writing, and will serve as an indication of the popularity, accessibility and wide readership of Loane's social studies