This paper presents the G-network, a new topological design which is a suitable architecture for point-to-point communication and interconnection networks, We show that the G-network has the following desirable characteristics: Efficient routing, small number of links, and fault tolerance. The performance of the G-network is compared to that of the Barrel Shifter and Illiac Mesh networks.

A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The minimum cardinality of a paired dominating set of G is the paired domination number of G. This chapter presents a survey of the major results on paired domination with an emphasis on bounds for the paired domination number.

A coalition in a graph (Formula presented.) consists of two disjoint sets of vertices V 1 and V 2, neither of which is a dominating set but whose union (Formula presented.) is a dominating set. A coalition partition in a graph G of order (Formula presented.) is a vertex partition (Formula presented.) such that every set Vi of π either is a dominating set consisting of a single vertex of degree n–1, or is not a dominating set but forms a coalition with another set (Formula presented.) which is not a dominating set. In this paper we introduce this concept and study its properties.

This article is about the alternative forms of power emerging in contemporary activism. It conceptualizes this new form of power as ‘non-dominating’, and puts forward six propositions which characterize this form of power. It builds on work about power with eight diverse communities in the North of England, to argue that this form of power does exist in practice at the neighbourhood level, even though it is not articulated as such. While neighbourhood activists have difficulty in making this form of power effective, at the level of the ‘square’ and global activism, new understandings and practices of power are under conscious experimentation. This contribution therefore suggests that better connections need to be built between these levels of activism. At the same time, non-dominating power should be recognized as enhancing the debate about effective and transformative change and how it can avoid reproducing dominating power.

An odd open dominating set of a graph is a subset of the graph’s vertices with the property that the open neighborhood of each vertex in the graph contains an odd number of vertices in the subset. An odd closed r-dominating set is a subset of the graph’s vertices with the property that the closed r-ball centered at each vertex in the graph contains an odd number of vertices in the subset. We first prove that the n-fold direct product of simple graphs has an odd open dominating set if and only if each factor has an odd open dominating set. Secondly, we prove that the n-fold strong product of simple graphs has an odd closed r-dominating set if and only if each factor has an odd closed r-dominating set.

Thesis--Auburn University, 2008. / Abstract. Vita. Includes bibliographical references (p. 45-48).

Understanding RNA molecules is important to genomics research. Recently researchers at the Courant Institute of Mathematical Sciences used graph theory to model RNA molecules and provided a database of trees representing possible secondary RNA structures. In this thesis we use domination parameters to predict which trees are more likely to exist in nature as RNA structures. This approach appears to have promise in graph theory applications in genomics research.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2014-08-12T15:13:32Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) dissertacao rommel cc.pdf: 1665635 bytes, checksum: 9f894f847272036c011387e2de71507f (MD5) / Made available in DSpace on 2014-08-12T15:13:32Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) dissertacao rommel cc.pdf: 1665635 bytes, checksum: 9f894f847272036c011387e2de71507f (MD5) Previous issue date: 2009-03-12 / Given a graph G = (V;E) and a set of vertices D V, a vertice v 2 V is dominated by D if jN[v] \ Dj 1. When jN(v) \ Dj = 1 for all v 2 V, G is efficiently dominable. A generalization of this concept is called efficient multiple domination, which requires all vertices must be dominated by a set D V exactly k times. The aim of this dissertation is to study these topics, describing the theoretical knowledge needed for advanced researches. For this reason, many of the theorems and its proofs are detailed. Furthermore, some results on the efficient multiple domination are presented, including bounds for the size of efficient k-dominating sets, the complement and iterated line graphs of efficiently (r + 1)-dominable r-regular graphs and a N P-completeness proof for the efficient multiple domination problem in arbitrary graphs. It is expected that this work contribute to the development of future researches on the efficient domination and in the resolution of some open problems. / Dado um grafo G = (V;E) e um subconjunto de vértices D V, define-se D como um conjunto dominante de G se todo vértice v 2 V que não estiver incluído no conjunto D for adjacente a pelo menos um vértice de D. Na situação em que, para todo v 2 V, jN[v]\Dj = 1, diz-se que o grafo G é eficientemente dominado. Uma generalização desse conceito consiste na múltipla dominação eficiente, em que é requerido que todo vértice do grafo seja dominado exatamente k vezes. O objetivo deste trabalho é realizar um estudo exploratório sobre esses temas, de modo a reunir o conhecimento teórico requerido para pesquisas avançadas. Para isso, buscou-se a apresentação e o detalhamento das demonstrações dos teoremas estudados. Além disso, foram fornecidos alguns resultados sobre a múltipla dominação eficiente no que se refere aos limites para o tamanho de um conjunto k-dominante eficiente, à relação da k-dominação eficiente entre grafos regulares, seu complemento e seus grafos linha iterados, bem como à caracterização da N P-completude para o problema da múltipla dominação eficiente em grafos arbitrários. Espera-se que esta dissertação forneça subsídios teóricos para estudos futuros voltados à dominação eficiente, bem como à resolução de algumas questões em aberto.

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