Are there more almost separable partitions than separable partitions?

A partition of a set of n points in d-dimensional space into p parts is called an (almost) separable partition if the convex hulls formed by the parts are (almost) pairwise disjoint. These two partition classes are the most encountered ones in clustering and other partition problems for high-dimensional points and their usefulness depends critically on the issue whether their numbers are small. The problem of bounding separable partitions has been well studied in the literature (Alon and Onn in Discrete Appl. Math. 91:39–51, 1999; Barnes et al. in Math. Program. 54:69–86, 1992; Harding in Proc. Edinb. Math. Soc. 15:285–289, 1967; Hwang et al. in SIAM J. Optim. 10:70–81, 1999; Hwang and Rothblum in J. Comb. Optim. 21:423–433, 2011a). In this paper, we prove that for d≤2 or p≤2, the maximum number of almost separable partitions is equal to the maximum number of separable partitions.     

Generalized perfect domination in graphs

Let k be a positive integer and G=(V,E) be a graph. A vertex subset D of a graph G is called a perfect k-dominating set of G, if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γ _kp (G). In this paper, we give characterizations of graphs for which γ _kp (G)=γ(G)+k?2 and prove that the perfect k-domination problem is NP-complete even when restricted to bipartite graphs and chordal graphs. Also, by using dynamic programming techniques, we obtain an algorithm to determine the perfect k-domination number of trees.     

Current Constraints and Future Directions for Risk Adapted Land-Use Planning Practices in the High-Density Asian Setting of Ho Chi Minh City

The need to integrate adaptation efforts into land-use planning policies has been only recently recognized in Ho Chi Minh City. The city's latest planning guidance addresses both flooding resilience and mitigation of urban heat. This paper outlines the development contexts and the current barriers for adapted land-use planning within the city. The key challenge for land-use planning is communicating the important functions and services of open and natural urban spaces and effectively guiding the mainly individual developer-driven development. As the realization of non-structural adaptation measures is in strong contrast to the current market-driven private and short-term developer interests, the main development trajectories are questioned and synergies identified.     

Developmental Correlates of Family Size: A Report from the National Child Development Study

Summary This paper presents findings from tbe National Child DevelopmentStudy which examine the relationship between size of familyand school attainment, adjustment and height at the age of eleven.This relationship is shown to be a strong one, and alternativeexplanations for its existence are considered