An improved upper bound for the bondage number of graphs on surfaces

The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. Recently Gagarin and Zverovich showed that, for a graph $G$ with maximum degree $\Delta(G)$ and embeddable on an orientable surface of genus $h$ and a non-orientable surface of genus $k$, $b(G)\leq\min\{\Delta(G)+h+2,\Delta+k+1\}$. They also gave examples showing that adjustments of their proofs implicitly provide better results for larger values of $h$ and $k$. In this paper we establish an improved explicit upper bound for $b(G)$, using the Euler characteristic $\chi$ instead of the genera $h$ and $k$, with the relations $\chi=2-2h$ and $\chi=2-k$. We show that $b(G)\leq\Delta(G)+\lfloor r\rfloor$ for the case $\chi\leq0$ (i.e. $h\geq1$ or $k\geq2$), where $r$ is the largest real root of the cubic equation $z^3+2z^2+(6\chi-7)z+18\chi-24=0$. Our proof is based on the technique developed by Carlson-Develin and Gagarin-Zverovich, and includes some elementary calculus as a new ingredient. We also find an asymptotically equivalent result $b(G)\leq\Delta(G)+\lceil\sqrt{12-6\chi\,}-1/2\rceil$ for $\chi\leq0$, and a further improvement for graphs with large girth.Comment: 8 pages, to appear in Discrete Mathematic

Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

Heredity for generalized power domination

In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for $\gamma\_{p,k}(G-e)$, $\gamma\_{p,k}(G/e)$ and for $\gamma\_{p,k}(G-v)$ in terms of $\gamma\_{p,k}(G)$, and give examples for which these bounds are tight. We characterize all graphs for which $\gamma\_{p,k}(G-e) = \gamma\_{p,k}(G)+1$ for any edge $e$. We also consider the behaviour of the propagation radius of graphs by similar modifications.Comment: Discrete Mathematics and Theoretical Computer Science, 201

An empirical examination of the three elements (actions, means and purpose) of the Palermo Protocol to establish an offence of human trafficking

The internationally agreed definition of human trafficking, contained in the Protocol to Prevent, Suppress and Punish Trafficking in Persons Especially Women and Children (Palermo Protocol), supplementing the United Nations Convention against Transnational Organized Crime is comprised of three elements: action, means and purpose. Empirical exploratory research considers the extent to which the definitional construct of three elements reflects convicted offender method to commit human trafficking. Empirical research was conducted on 972 offenders convicted of human trafficking and the actions and means they used to fulfil different purposes to commit human trafficking. Data was collected and disaggregated from 486 conviction case summaries contained in SHERLOC, the United Nations Office on Drugs & Crime database, related to prosecutions brought by 40 Member States to the Palermo Protocol. Analysis explores academic discord on the extent to which human trafficking is the process of moving a victim to the point of exploitation, but not including exploitation of the victim (Chuang, 2014) (Stoyanova, 2015a) or includes both the process of moving the victim and the static action of end exploitation (Gallagher, 2010). Furthermore, empirical analysis is made of the actual actions and means performed by offenders to further an understanding of problematic terms in the definition and explore other insights from an analysis of the three elements. Finally, empirical analysis through structural equation modelling explores an order and structure to human trafficking and results are presented through a series of visuals to facilitate the practical translation of findings for investigators

Relations of Production and Modes of Surplus Extraction in India

This paper uses aggregate-level data, as well as case-studies, to trace out the evolution of some key structural features of the Indian economy, relating both to the agricultural and the informal industrial sector. These aggregate trends are used to infer: (a) the dominant relations of production under which the vast majority of the Indian working people labour, and (b) the predominant ways in which the surplus labour of the direct producers is appropriated by the dominant classes. This summary account is meant to inform and link up with on-going attempts at radically restructuring Indian society. JEL Categories: B24, B51relations of production, forms of surplus extraction, mode of production, India

A Reexamination of El Greco’s View and Plan of Toledo as a Question of Sources and Patronage

El Greco’s View and Plan of Toledo (c. 1610, Museo del Greco, Toledo) explores multiple ways of representing space by juxtaposing a sweeping view of the city of Toledo with a trompe l’oeil map of the city’s streets. Recent scholarship has shown that that El Greco probably copied the map from a plan of Toledo in the Atlas de El Escorial (1538-45), a royal commission that would have been the first complete atlas of Spain. Significantly, this atlas was likely available to the owner and probable patron of View and Plan of Toledo, the scholar and map collector Pedro Salazar de Mendoza. Although art historians have often seen View and Plan of Toledo as an expression of El Greco’s singular, “self-conscious” skill as a painter and draughtsman, I argue the painting should be read as an intellectual collaboration between the artist and Salazar. In the painting, El Greco gave pictorial form to Toledan geography, theology, history and law, themes that are likewise reflected in Pedro Salazar’s writings and post-mortem inventories. I draw from various seventeenth-century images and texts, including El Greco’s body of work, the work of Cretan icon painters, maps and books from Salazar’s collection, and texts and treatises circulating in the seventeenth century to re-orient contemporary scholarship on the painting and re-illuminate this enigmatic cityscape