Vertex-Coloring with Star-Defects

Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, and acyclicity of the monochromatic components. In this paper we focus on defective colorings in which the monochromatic components are acyclic and have small diameter, namely, they form stars. For outerplanar graphs, we give a linear-time algorithm to decide if such a defective coloring exists with two colors and, in the positive case, to construct one. Also, we prove that an outerpath (i.e., an outerplanar graph whose weak-dual is a path) always admits such a two-coloring. Finally, we present NP-completeness results for non-planar and planar graphs of bounded degree for the cases of two and three colors

Gender based water violence: cross cultural evidence of severe harm associated with water insecurity for women and girls

We examined how study participants in Indonesia and Peru viewed the relationship between water insecurity and women's health via thematic analysis of interviews and focus groups. Participants reported that water insecurity led to vaginal infections, miscarriage, premature births, uterine prolapse, poor nutrition, restricted economic opportunities, and intergenerational cycles of poverty. Participants in both countries stated that extreme burdens associated with water insecurity should be categorized as violence. Based on these findings, we developed the concept of “gender-based water violence,” defined as the spectrum of stressors associated with water insecurity that are so severe as to threaten human health and well-being, particularly that of women and girls

Irreducible triangulations of surfaces with boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary

A topological classification of convex bodies

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale functions on S^2 by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in M_C (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph P_2 and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity-preserving local manipulation of the surface. Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist, this algorithm not only proves our claim but also generalizes the known classification scheme in [36]. Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. in [21], producing a hierarchy of increasingly coarse Morse-Smale complexes. We point out applications to pebble shapes.Comment: 25 pages, 10 figure

Local chromatic number of quadrangulations of surfaces

The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number

Extending Torelli map to toroidal compactifications of Siegel space

It has been known since the 1970s that the Torelli map $M_g \to A_g$, associating to a smooth curve its jacobian, extends to a regular map from the Deligne-Mumford compactification $\bar{M}_g$ to the 2nd Voronoi compactification $\bar{A}_g^{vor}$. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification $\bar{A}_g^{perf}$ is also regular, and moreover $\bar{A}_g^{vor}$ and $\bar{A}_g^{perf}$ share a common Zariski open neighborhood of the image of $\bar{M}_g$. We also show that the map to the Igusa monoidal transform (central cone compactification) is NOT regular for $g\ge9$; this disproves a 1973 conjecture of Namikawa.Comment: To appear in Inventiones Mathematica

Special Feature CKD as a Model for Improving Chronic Disease Care through Electronic Health Records

Abstract Electronic health records have the potential to improve the care of patients with chronic medical conditions. CKD provides a unique opportunity to show this potential: the disease is common in the United States, there is significant room to improve CKD detection and management, CKD and its related conditions are defined primarily by objective laboratory data, CKD care requires collaboration by a diverse team of health care professionals, and improved access to CKD-related data would enable identification of a group of patients at high risk for multiple adverse outcomes. However, to realize the potential for improvement in CKD-related care, electronic health records will need to provide optimal functionality for providers and patients and interoperability across multiple health care settings. The goal of the National Kidney Disease Education Program Health Information Technology Working Group is to enable and support the widespread interoperability of data related to kidney health among health care software applications to optimize CKD detection and management. Over the course of the last 2 years, group members met to identify general strategies for using electronic health records to improve care for patients with CKD. This paper discusses these strategies and provides general goals for appropriate incorporation of CKD-related data into electronic health records and corresponding design features that may facilitate (1) optimal care of individual patients with CKD through improved access to clinical information and decision support, (2) clinical quality improvement through enhanced population management capabilities, (3) CKD surveillance to improve public health through wider availability of population-level CKD data, and (4) research to improve CKD management practices through efficiencies in study recruitment and data collection. Although these strategies may be most effectively applied in the setting of CKD, because it is primarily defined by laboratory abnormalities and therefore, an ideal computable electronic health record phenotype, they may also apply to other chronic diseases