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We introduce and study a constrained planarity testing problem, called 1 -Fixed Constrained Planarity , and prove that this problem can be solved in quadratic time for biconnected graphs. Our solution is based on a novel definition of fixedness that makes it possible to simplify and extend known techniques about Simultaneous PQ-Ordering. We exploit this result to study different versions of the hybrid planarity testing problem. Namely, we show polynomial-time solutions for a variant of NodeTrix Planarity with fixed sides, for PolyLink Planarity , and for Clique Planarity with fixed sides. [ABSTRACT FROM AUTHOR]

An ortho-polygon visibility representation (OPVR) of an embedded graph G is an embedding-preserving drawing that maps each vertex of G to a distinct orthogonal polygon and each edge of G to a vertical or horizontal visibility between its end-vertices. An OPVR Γ has vertex complexity k if every polygon of Γ has at most k reflex corners. A 1-plane graph is an embedded graph such that each edge is crossed at most once. It is known that 3-connected 1-plane graphs admit an OPVR with vertex complexity at most 12, while vertex complexity at least 2 may be required in some cases. In this paper, we reduce this gap by showing that vertex complexity 5 is always sufficient, while vertex complexity 4 may be sometimes required. These results are based on the study of the combinatorial properties of the B-, T-, and W-configurations in 3-connected 1-plane graphs. An implication of the upper bound is the existence of a O ˜ (n 10 7 ) -time drawing algorithm that computes an OPVR of an n -vertex 3-connected 1-plane graph on an integer grid of size O (n) × O (n) and with vertex complexity at most 5. [ABSTRACT FROM AUTHOR]

A subcubic planar graph is a planar graph whose vertices have degree at most 3. We show that the subcubic planar graphs with at least five vertices have planar slope number at most 4, which is worst-case optimal. This answers an open question by Jelínek et al. [10] . Furthermore, we prove that the subcubic planar graphs with at least five vertices have angular resolution π / 4 , which solves an open problem by Kant [11] and by Formann et al. [8] . [ABSTRACT FROM AUTHOR]

The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge. The study of 1-planar graphs dates back to more than fifty years ago and, recently, it has driven increasing attention in the areas of graph theory, graph algorithms, graph drawing, and computational geometry. This annotated bibliography aims to provide a guiding reference to researchers who want to have an overview of the large body of literature about 1-planar graphs. It reviews the current literature covering various research streams about 1-planarity, such as characterization and recognition, combinatorial properties, and geometric representations. As an additional contribution, we offer a list of open problems on 1-planar graphs. [ABSTRACT FROM AUTHOR]

A 1-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A 1-plane graph is optimal if it has maximum edge density. A red–blue edge coloring of an optimal 1-plane graph G partitions the edge set of G into blue edges and red edges such that no two blue edges cross each other and no two red edges cross each other. We prove the following: ( i ) Every optimal 1-plane graph has a red–blue edge coloring such that the blue subgraph is maximal planar while the red subgraph has vertex degree at most four; this bound on the vertex degree is worst-case optimal. ( ii ) A red–blue edge coloring may not always induce a red forest of bounded vertex degree. Applications of these results to graph augmentation and graph drawing are also discussed. [ABSTRACT FROM AUTHOR]

Purpose of the study The prevalence of frailty is expected to increase worldwide in parallel with demographic ageing. Despite this, little is known about the prevalence in different populations particularly community-based samples. This cross-sectional study evaluates the prevalence of frailty in a community-dwelling older adult population and describes a methodology to plan community-based interventions. Methodology A random sample of 1331 older adults, resident in the Lazio-Region of Italy, were screened by trained public health nurses (PHNs) by administering a validated questionnaire (the Functional Geriatric Evaluation questionnaire). Prevalence of frailty was calculated using the Final Synthetic Score derived from the questionnaire’s Final Score. Variables associated with frailty were selected through univariate and multivariate statistical analysis. Results Prevalence of frail (FS ≥ 10,≤50) and very frail (FS < 10) individuals was 13.9% and 7.6% respectively. Variables associated with frailty were age (older than 85 years), disability, living alone or the presence of a paid carer, lower education and neurological disorders like stroke, dementia, Parkinson disease and other neuropsychiatric diseases; Anaemia or cancer were also associated with a higher prevalence of frailty. Discussion The study provide a comprehensive picture of the prevalence of frailty and factors associated to this condition in community-dwelling older adults. On the basis of the study results, a plan of community-based services could address the needs of care of the elderly population. A trained team of PHNs may be the most appropriate personnel to carry out multidimensional frailty assessment in this setting. [ABSTRACT FROM AUTHOR]

Let 〈 G r , G b 〉 be a pair of plane st -graphs with the same vertex set V . A simultaneous visibility representation with L-shapes of 〈 G r , G b 〉 is a pair of bar visibility representations 〈 Γ r , Γ b 〉 such that, for every vertex v ∈ V , Γ r ( v ) and Γ b ( v ) are a horizontal and a vertical segment, respectively, which share an end-point. In other words, every vertex is drawn as an L-shape, every edge of G r is a vertical visibility segment, and every edge of G b is a horizontal visibility segment. Also, no two L-shapes intersect each other. An L-shape has four possible rotations, and we assume that each vertex is given a rotation for its L-shape as part of the input. Our main results are: (i) a characterization of those pairs of plane st -graphs admitting such a representation, (ii) a quadratic time algorithm to recognize them, and (iii) a linear time drawing algorithm if the test is positive. As an application, starting from a simultaneous visibility representation with L-shapes, we show how to compute a simultaneous embedding of the two graphs with at most two bends per edge and right-angle crossings. [ABSTRACT FROM AUTHOR]

Abstract: Let R be a set of red points and let B be a set of blue points. The point set is called equitable if and it is called general if no two points are vertically or horizontally aligned. An orthogeodesic alternating path on P is a path such that each edge is an orthogeodesic chain connecting points of different color and such that no two edges cross. We consider the problem of deciding whether a set of red and blue points admits a Hamiltonian orthogeodesic alternating path, that is, an orthogeodesic alternating path visiting all points. We prove that every general equitable point set admits a Hamiltonian orthogeodesic alternating path and we present an -time algorithm for finding such a path, where n is the number of points. On the other hand, we show that the problem is -complete if the path must be on the grid (i.e., vertices and bends have integer coordinates). Further, we show that we can approximate the maximum length of an orthogeodesic alternating path on the grid by a factor of 3, whereas we present a family of point sets with n points that do not have a Hamiltonian orthogeodesic alternating path with more than points. Additionally, we show that it is -complete to decide whether a given set of red and blue points on the grid admits an orthogeodesic perfect matching if horizontally aligned points are allowed. This contrasts a recent result by Kano (2009) who showed that this is possible on every general point set. [Copyright &y& Elsevier]

Abstract: This paper studies how to compute radially layered drawings of graphs by taking into account additional geometric constraints which correspond to typical aesthetic and semantic requirements for the visualization. The following requirements are considered: vertex centrality, edge crossings, curve complexity, and radial distribution of the vertices. Trade-offs among these requirements are discussed and different linear-time drawing algorithms are presented. [Copyright &y& Elsevier]

Abstract: This paper studies weak proximity drawings of graphs and demonstrates their advantages over strong proximity drawings in certain cases. Weak proximity drawings are straight line drawings such that if the proximity region of two points p and q representing vertices is devoid of other points representing vertices, then segment is allowed, but not forced, to appear in the drawing. This differs from the usual, strong, notion of proximity drawing in which such segments must appear in the drawing. Most previously studied proximity regions are associated with a parameter β, . For fixed β, weak β-drawability is at least as expressive as strong β-drawability, as a strong β-drawing is also a weak one. We give examples of graph families and β values where the two notions coincide, and a situation in which it is NP-hard to determine weak β-drawability. On the other hand, we give situations where weak proximity significantly increases the expressive power of β-drawability: we show that every graph has, for all sufficiently small β, a weak β-proximity drawing that is computable in linear time, and we show that every tree has, for every β less than 2, a weak β-drawing that is computable in linear time. [Copyright &y& Elsevier]

• For n ≥ 14 there is a properly 2-colored SP-graph with 2 n + 4 vertices s.t. any 2-colored PSE requires Ω (n) bends on Ω (n) edges • 2 bends per edge suffice for a 2-colored PSE of properly 2-colored outerplanar graphs • 1 bend per edge suffices for a 2-colored PSE of a properly 2-colored outerplanar graph if the points are linearly separable • 3 bends per edge suffice for a 2-colored PSE of a 2-colored outerplanar graph • 1 bend per edge suffices for a 2-colored PSE of a 2-colored caterpillar Let G be a planar graph whose vertices are colored either red or blue and let S be a set of points having as many red (resp. blue) points as the red (resp. blue) vertices of G. A 2-colored point-set embedding of G on S is a planar drawing that maps each red (resp. blue) vertex of G to a red (resp. blue) point of S. We show that there exist partial 2-trees that are properly 2-colored (i.e., they are 2-colored with no two adjacent vertices have the same color), whose point-set embeddings may require linearly many bends on linearly many edges. For a contrast, we show that two bends per edge are sufficient for 2-colored point-set embedding of properly 2-colored outerplanar graphs. For separable point sets this bound reduces to one, which is worst-case optimal. If the 2-coloring of the outerplanar graph is not proper, three bends per edge are sufficient and one bend per edge (which is worst-case optimal) is sufficient for caterpillars. [ABSTRACT FROM AUTHOR]

We present a new model for hybrid planarity that relaxes existing hybrid representation models. A graph G = (V , E) is (k , p) -planar if V can be partitioned into clusters of size at most k such that G admits a drawing where: (i) each cluster is associated with a closed, bounded planar region, called a cluster region ; (ii) cluster regions are pairwise disjoint, (iii) each vertex v ∈ V is identified with at most p distinct points, called ports , on the boundary of its cluster region; (iv) each inter-cluster edge (u , v) ∈ E is identified with a Jordan arc connecting a port of u to a port of v ; (v) inter-cluster edges do not cross or intersect cluster regions except at their end-points. We first tightly bound the number of edges in a (k , p) -planar graph with p < k. We then prove that (4 , 1) -planarity testing and (2 , 2) -planarity testing are NP-complete problems. Finally, we prove that neither the class of (2 , 2) -planar graphs nor the class of 1-planar graphs contains the other, indicating that the (k , p) -planar graphs are a large and novel class. [ABSTRACT FROM AUTHOR]

We establish new results on the curve complexity of k -colored point-set embeddings when k = 3. We show that there exist 3-colored caterpillars with only three independent edges whose 3-colored point-set embeddings may require Ω (n 1 3 ) bends on Ω (n 2 3 ) edges. This settles an open problem by Badent et al. [5] about the curve complexity of point set embeddings of k -colored trees and it extends a lower bound by Pach and Wenger [35] to the case that the graph only has O (1) independent edges. Concerning upper bounds, we prove that any 3-colored path admits a 3-colored point-set embedding with curve complexity at most 4. In addition, we introduce a variant of the k -colored simultaneous embeddability problem and study its relationship with the k -colored point-set embeddability problem. [ABSTRACT FROM AUTHOR]

We study the problem of representing topological graphs as polyline drawings with few bends per edge and such that the topology of the graph is either fully or partially preserved. More formally, let G be a simple topological graph and let Γ be a polyline drawing of G. Drawing Γ partially preserves the topology of G if it has the same external boundary, the same circular order of the edges around each vertex, and the same set of crossings as G , while it fully preserves the topology of G if the planarization of G and the planarization of Γ have the same planar embedding. We prove that if the set of crossing-free edges of G forms a biconnected (connected) spanning subgraph, then G admits a polyline drawing that partially preserves its topology and that has curve complexity at most one (three), i.e., with at most one (three) bend(s) per edge. If, however, the set of crossing-free edges of G is not a connected spanning subgraph, the curve complexity may be Ω (n) , while it is O (1) if the number of connected components is O (1). Concerning drawings that fully preserve the topology, we show that if G is k -skew (i.e., it becomes planar after removing k suitably chosen edges), it admits one such drawing with curve complexity at most 2 k ; for 1-skew graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal 2-plane graphs (i.e., with at most two crossings per edge and maximum edge density), for which we discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology. [ABSTRACT FROM AUTHOR]

• Hypergammaglobulinemia is common in Malawian HIV-positive pregnant women. • Twenty-four months of antiretroviral therapy improved but did not normalize IgG anomalies. • The IgG2 isotype remained highly underrepresented. • The potential influence of immunoglobulin disorders should be considered when assessing the risk of infection in HIV-exposed infants. Hypergammaglobulinemia and anomalies in the IgG subclass distribution are common in HIV-infected individuals and persist even after many years of antiretroviral therapy (ART). The aim of this study was to investigate the IgG profile and dynamics in pregnant HIV-infected Malawian women in the Option B era. Thirty-seven treatment-naive women received ART from the third trimester of pregnancy to 6 months post delivery (end of the breastfeeding period). ART continuation (group C) or interruption (group I) was then decided on the basis of the CD4+ cell count at enrolment (>350 or ≤350/μl). Total IgG and IgG subclasses were determined in maternal serum using a nephelometric assay at baseline and at 6 and 24 months postpartum. At enrolment, 36/37 women had IgG levels >15 g/l and there was a predominance of the IgG1 isotype (more than 90%) in parallel with underrepresentation of IgG2 (5.0%). After 6 months of ART, both groups showed a significant median decrease in total IgG (−3.1 g/l in group I, −3.5 g/l in group C) and in IgG1 (−4.0 g/l and −3.6 g/l, respectively), but only a modest recovery in IgG2 levels (+0.16 in group I, +0.14 g/l in group C). At month 24, hypergammaglobulinemia was still present in 73.7% of women in group C, although a significant reduction was observed in total IgG level and in IgG1 and IgG3 subclasses (p < 0.0001 in all cases). IgG2 levels did not show any significant change. In group I at 24 months, total IgG and IgG subclasses had returned to levels comparable to those at baseline. The beneficial effects of 24 months of ART appear to be limited in the B-cell compartment, with an incomplete reduction of total IgG levels and no recovery of IgG2 depletion. A short ART period did not have significant effects on IgG abnormalities in women who interrupted treatment. [ABSTRACT FROM AUTHOR]

Abstract We show that any outerplanar graph admits a planar straight-line drawing such that the length ratio of the longest to the shortest edges is strictly less than 2. This result is tight in the sense that for any ϵ > 0 there are outerplanar graphs that cannot be drawn with an edge-length ratio smaller than 2 − ϵ. We also show that this ratio cannot be bounded if the embeddings of the outerplanar graphs are given. [ABSTRACT FROM AUTHOR]