Disjunctive total domination in graphs

Let \(G\) be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, \(\gamma _t(G)\). A set \(S\) of vertices in \(G\) is a disjunctive total dominating set of \(G\) if every vertex is adjacent to a vertex of \(S\) or has at least two vertices in \(S\) at distance \(2\) from it. The disjunctive total domination number, \(\gamma ^d_t(G)\), is the minimum cardinality of such a set. We observe that \(\gamma ^d_t(G) \le \gamma _t(G)\). We prove that if \(G\) is a connected graph of order \(n \ge 8\), then \(\gamma ^d_t(G) \le 2(n-1)/3\) and we characterize the extremal graphs. It is known that if \(G\) is a connected claw-free graph of order \(n\), then \(\gamma _t(G) \le 2n/3\) and this upper bound is tight for arbitrarily large \(n\). We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if \(G\) is a connected claw-free graph of order \(n > 14\), then \(\gamma ^d_t(G) \le 4n/7\) and we characterize the graphs achieving equality in this bound.     

Service learning and the practice of social justice and care

The values of a social justice pedagogy in teacher education are assumed to be firmly established in South African higher education. This article discusses how serviced learning (SL) can provide practical experience of caring and serve the promotion of the ideals of social justice in two initial teacher education programmes. However, an analysis of data in this case study shows that, although SL practice has served as a platform for social justice and for care as two basic educational values, participants were not yet able to verbalize and theorize these values. They were also not able to embed these values in their reflection about service as an inherent component of education. With data from different role players in a number of SL projects, we show that the students’ discourse of what it means to be a caring teacher in a just society has not yet been developed despite the successful practical experience. It was found that, although the students had been building a practice ‘platform’ for service, they had not yet been able to articulate the conceptual intersect of care and social justice clearly. With this epistemological notion in mind, two main themes from the data analysis are discussed: 1) How SL shifted student learning from ideas ‘about’ service to implemented service, and 2) how the practical experience promoted reciprocity in inter-institutional and inter-generational communities of practice.     

Constraint bipartite vertex cover: simpler exact algorithms and implementations

Constraint bipartite vertex cover is a graph-theoretical formalization of the spare allocation problem for reconfigurable arrays. We report on an implementation of a parameterized algorithm for this problem. This has led to considerable simplifications of the published, quite sophisticated algorithm. Moreover, we can prove that the mentioned algorithm could be quite efficient in practial situations. We also discuss possible generalizations and enhancements.     

Edge lifting and total domination in graphs

Let u and v be vertices of a graph G, such that the distance between u and v is two and x is a common neighbor of u and v. We define the edge lift of uv off x as the process of removing edges ux and vx while adding the edge uv to G. In this paper, we investigate the effect that edge lifting has on the total domination number of a graph. Among other results, we show that there are no trees for which every possible edge lift decreases the total domination number and that there are no trees for which every possible edge lift leaves the total domination number unchanged. Trees for which every possible edge lift increases the total domination number are characterized.     

Crown width models for woody plant species growing in urban areas of the U.S.

Urban Ecosystems - Crown widths of woody plant species growing in urban areas are of considerable importance as an overall indicator of health and also serve as an important factor for assessing...     

Intercultural Learning in a Multicultural Society

In this contribution the author describes intercultural education as education for higher order skills, that go beyond traditional teaching methods. Intercultural education emphasises interactive skills, it strengthens the self‐esteem of children by giving them a solid foundation to encounter the unknown.     

Connecting sleep,the neurocognitive memory system,and Bourdieu's habitus concept: Is sleep a generative force of the habitus?

Bourdieu's habitus theory has been criticized for allegedly being static and deterministic. This is partly due to insufficiently elaborated microstructural aspects of the habitus. Previous works have tried to elaborate the habitus' individual side by drawing on neuropsychological and neuroscientific concepts including the neuropsychological memory system. Despite its relevance for several social processes, sleep has not been considered in this context. The present work investigates the interrelatedness of habitus, sleep, and the memory system with particular focus on the determinism and staticity critique. It shows that the neuropsychological memory system overlaps largely with implicit memory and sleep plays an important role in the formation of habitus‐relevant memory. Moreover, sleep enhances processes such as memory consolidation, reflexivity, associative creativity, and dream‐associated simulation of unknown situations which counter the determinism and staticity critique. The present article argues that habitus, memory, and sleep are closely related and memory and sleep represent operational micro‐structures of the habitus. Moreover, sleep‐related processes represent a generative force that renders the habitus more dynamic. Finally, strengths and weaknesses of the habitus‐memory‐sleep nexus are discussed and further lines of research are outlined.     

A short proof of a result on a Vizing-like problem for integer total domination

In this note, we provide a short proof of a recent result on a Vizing-like problem for integer total domination in graphs given by Li and Hou (J. Comb. Optim., 2008).