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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]

Let Γ be a straight-line drawing of a graph and let u and v be two vertices of Γ. The Gabriel disk of u , v is the disk having u and v as antipodal points. A pair 〈 Γ 0 , Γ 1 〉 of vertex-disjoint straight-line drawings forms a mutual witness Gabriel drawing when, for i = 0 , 1 , any two vertices u and v of Γ i are adjacent if and only if their Gabriel disk does not contain any vertex of Γ 1 − i. We characterize the pairs 〈 G 0 , G 1 〉 of complete bipartite graphs that admit a mutual witness Gabriel drawing. The characterization leads to a linear time testing algorithm. We also show that when the pair 〈 G 0 , G 1 〉 consists of two complete multi-partite graphs whose partition sets all have size greater than one, then the pair does not admit a mutual witness Gabriel drawing unless the pair is 〈 K 2 , 2 , K 2 , 2 〉. [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]

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]

Let G be a 3-connected planar graph with n vertices and let p ( G ) be the maximum number of vertices of an induced subgraph of G that is a path. Substantially improving previous results, we prove that p ( G ) ≥ log ⁡ n 12 log ⁡ log ⁡ n . To demonstrate the tightness of this bound, we notice that the above inequality implies p ( G ) ∈ Ω ( ( log 2 ⁡ n ) 1 − ε ) , where ε is any positive constant smaller than 1, and describe an infinite family of planar graphs for which p ( G ) ∈ O ( log ⁡ n ) . As a byproduct of our research, we prove a result of independent interest: Every 3-connected planar graph with n vertices contains an induced subgraph that is outerplanar and connected and that contains at least n 3 vertices. The proofs in the paper are constructive and give rise to O ( n ) -time algorithms. [ABSTRACT FROM AUTHOR]

Abstract: A Right Angle Crossing Graph (also called a RAC graph for short) is a graph that has a straight-line drawing where any two crossing edges are orthogonal to each other. A -planar graph is a graph that has a drawing where every edge is crossed at most once. This paper studies the combinatorial relationship between the family of RAC graphs and the family of -planar graphs. It is proved that: (1) all RAC graphs having maximal edge density belong to the intersection of the two families; and (2) there is no inclusion relationship between the two families. As a by-product of the proof technique, it is also shown that every RAC graph with maximal edge density is the union of two maximal planar graphs. [Copyright &y& Elsevier]

Abstract: This paper studies the problem of drawing directed acyclic graphs in three dimensions in the straight-line grid model so that all directed edges are oriented in a common (upward) direction. We show that there exists a family of outerplanar directed acyclic graphs whose volume requirement is super-linear. We also prove that for the case of directed trees a linear-volume upper bound is achievable. [Copyright &y& Elsevier]

Abstract: This paper investigates the basic problem of computing crossing-free straight-line 3D grid drawings of graphs such that the overall volume is small. Motivated by their relevance in the literature, we focus on families of graphs having constant queue number and on k-trees. We present algorithms that compute drawings of these families of graphs on 3D grids consisting of a constant number of parallel lines and such that the overall volume is linear. Lower bounds on the number of such grid lines are also provided. Our results extend and improve similar ones already described in the literature. [Copyright &y& Elsevier]

Given a set of n points in the plane, any β-skeleton and [γ0,γ1]-graph can be computed in quadratic time. The presented algorithms are optimal for β values that are less than 1 and [γ0,γ1] values that result in non-planar graphs. We show a numerically robust algorithm that computes Gabriel graphs in quadratic time and degree 2. We finally show how a β-spectrum can be computed in optimal O(n2) time. [Copyright &y& Elsevier]

We study the problem of characterizing sets of points whose Voronoi diagrams are trees and if so, what are the combinatorial properties of these trees. The second part of the problem can be naturally turned into the following graph drawing question: Given a tree T, can one represent T so that the resulting drawing is a Voronoi diagram of some set of points? We investigate the problem both in the Euclidean and in the Manhattan metric. The major contributions of this paper are as follows.

We determine the reachability properties of the embeddings in R3 of a directed path, in the graph-theoretic sense, whose edges have each been assigned a desired direction (East, West, North, South, Up, or Down) but no length. We ask which points of R3 can be reached by the terminus of an embedding of such a path, by choosing appropriate positive lengths for the edges, if the embedded path starts at the origin, does not intersect itself, and respects the directions pre-assigned to its edges. This problem arises in the context of extending planar graph embedding techniques and VLSI rectilinear layout techniques from 2D to 3D. We give a combinatorial characterization of reachability that yields linear time recognition and layout algorithms. Finally, we extend our characterization to Rd, d>3. [Copyright &y& Elsevier]

It is proved that every series-parallel digraph whose maximum vertex degree is Δ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of Δ distinct slopes. The construction is worst-case optimal in terms of the number of slopes, and it gives rise to drawings with optimal angular resolution π Δ. A variant of the drawing algorithm is used to show that (non-directed) reduced series-parallel graphs and flat series-parallel graphs have a (non-upward) 1-bend planar drawing with ⌈ Δ 2 ⌉ distinct slopes if biconnected, and with ⌈ Δ 2 ⌉ + 1 distinct slopes if connected. [ABSTRACT FROM AUTHOR]

• 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]

A drawing of a graph is 1-planar if each edge is crossed at most once. A graph is 1-planar if it has a 1-planar drawing. A k-bend RAC (Right Angle Crossing) drawing of a graph is a polyline drawing where each edge has at most k bends and edges cross only at right angles. A graph is k-bend RAC if it has a k -bend RAC drawing. A 0-bend RAC graph (drawing) is also called a straight-line RAC graph (drawing) . The relationships between 1-planar and k -bend RAC graphs have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC and straight-line RAC graphs that are not 1-planar. The existence of 1-planar straight-line RAC drawings has been proven only for restricted families of 1-planar graphs. Two of the main questions still open are: ( i ) What is the complexity of deciding whether a graph has a drawing that is both 1-planar and straight-line RAC? ( i i ) Does every 1-planar graph have a drawing that is both 1-planar and 1-bend RAC? In this paper we answer these two questions. Namely, we prove an NP-hardness result for the first question, and we positively answer the second question by describing a drawing algorithm for 1-planar graphs. [ABSTRACT FROM AUTHOR]

Abstract: To determine the trend of elderly hospitalization rates in Italy, following the economic rationalization of health systems in Western countries, and to evaluate which alternatives to acute hospitalization have been developed during the period 1996–2006 an ecological observational study has been carried out. Data from the Italian Hospital-Discharge Registries (HDRs) of the years 1996, 2001, and 2006 have been analyzed in order to assess the variations among the elderly in terms of hospitalization rates, hospital stay, and bed rates. The results were compared with nursing home admission rates and home care offer. Relations among these variables were explored by univariate and multivariate analyses. Elderly hospital admission rates decreased in Italy from 324.2/1000 in 1996, to 258.7 in 2006. Mean hospital stay of elderly patients was 9.4 days in 2006, 9.5 in 2001 and 10.1 in 1996. A multivariate linear regression model was statistically significant in explaining the variations in hospitalization rates (F: 5.68; p =0.004; R 2 =0.77). The main determinants linked to such variations were the bed rate (β =0.67; p =0.004) and the hospital length of stay (β =−0.77; p =0.03). The analysis showed a reduction in hospitalization rates among the elderly, which was not counterbalanced by an increased offer of home care and/or nursing home services, but was mainly linked to a decrease in the supply of acute beds, with possible consequences on the quality of health care. [Copyright &y& Elsevier]

Background Virtually all HIV-infected women in sub-Saharan Africa have evidence of Cytomegalovirus (CMV) infection and levels of specific anti-CMV IgG have been suggested to represent more intense reactivation of subclinical infection. Studies have also shown direct influence of CMV on lymphocytes. Objective The aim of this study was to determine if levels of anti-CMV specific antibodies could impact on the immunological response to antiretroviral treatment (ART) in HIV-infected pregnant women. Study design CMV-specific IgG were measured in HIV-infected pregnant women at 26 weeks of gestation (before ART initiation). Women received ART until 6 months postpartum or indefinitely according to local guidelines at the time of the study. Immunological and virological responses were assessed 6 months and 24 months after delivery. Results A total of 81 women were studied. At baseline high levels (above the median) of specific IgG were associated to a low CD4+ cell count (P< 0.001), a high viral load (P = 0.003), and to an older age (P = 0.051). In a multivariate model adjusting for baseline CD4+ count, baseline viral load and age, the presence of low levels of CMV IgG was the only independent predictor of a a CD4+ count above 500/mm 3 24 months after delivery among women on continuous therapy. Conclusions In this cohort, levels of CVM IgG had a significant influence on the immunological response to ART, adding information to the known impact of CMV infection in the HIV-positive population, and underlining the need of new strategies to contain the infection. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study how to compute compact straight-line drawings of planar graphs with a limited number of crossings per edge. We prove that every outerplanar graph can be drawn in area using a sub-linear number of crossings per edge, and that for any given number , every outerplanar graph admits an area drawing with crossing per edge. The drawing algorithms run in linear time and can be also generalized to another meaningful sub-family of series–parallel graphs with bounded vertex-degree. [Copyright &y& Elsevier]

Abstract: Cognitive experiments show that humans can read graph drawings in which all edge crossings are at right angles equally well as they can read planar drawings; they also show that the readability of a drawing is heavily affected by the number of bends along the edges. A graph visualization whose edges can only cross perpendicularly is called a RAC (Right Angle Crossing) drawing. This paper initiates the study of combinatorial and algorithmic questions related to the problem of computing RAC drawings with few bends per edge. Namely, we study the interplay between number of bends per edge and total number of edges in RAC drawings. We establish upper and lower bounds on these quantities by considering two classical graph drawing scenarios: The one where the algorithm can choose the combinatorial embedding of the input graph and the one where this embedding is fixed. [Copyright &y& Elsevier]

Abstract: The contribution of this paper is twofold. It presents a new approach to the matched drawability problem of pairs of planar graphs and it provides four algorithms based on this approach for drawing the pairs , , and . Further, it initiates the study of the matched drawability of triples of planar graphs: it presents an algorithm to compute a matched drawing of a triple of cycles and an algorithm to compute a matched drawing of a caterpillar and two unlabeled level planar graphs. The results extend previous work on the subject and relate to existing literature about simultaneous embeddability and unlabeled level planarity. [Copyright &y& Elsevier]

Abstract: Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A geometric point-set embedding is a point-set embedding with no edge bends. This paper studies the following problem: The input is a set S of n points, a planar graph G with n vertices, and a geometric point-set embedding of a subgraph on a subset of S. The desired output is a point-set embedding of G on S that includes the given partial drawing of . We concentrate on trees and show how to compute the output in time in a real-RAM model and with at most edges with at most bends, where k is the number of vertices of the given subdrawing. We also prove that there are instances of the problem which require at least bends on edges. [Copyright &y& Elsevier]

Abstract: Let be a planar graph with vertices and with a partition of the vertex set into subsets for some positive integer . Let be a set of distinct points in the plane with a partition into subsets with (). This paper studies the problem of computing a planar polyline drawing of , such that each vertex of is mapped to a distinct point of . Lower and upper bounds on the number of bends per edge are proved for any . In the special case , we improve the upper and lower bounds presented in a paper by Pach and Wenger [J. Pach, R. Wenger, Embedding planar graphs at fixed vertex locations, Graphs and Combinatorics 17 (2001) 717–728]. The upper bound is based on an algorithm for computing a topological book embedding of a planar graph, such that the vertices follow a given left-to-right order and the number of crossings between every edge and the spine is asymptotically optimal, which can be regarded as a result of independent interest. [Copyright &y& Elsevier]

Abstract: In this paper, we study k-spine, h-bend planar drawings in which each vertex of a planar graph G lies on one of horizontal lines and each edge of G is drawn as a polyline containing at most bends. A graph with a k-spine, h-bend planar drawing is said to be k-spine, h-bend planar. We mainly focus on k-spine, 1-bend planar drawings, showing that for each , there exists a planar graph that is not k-spine, 1-bend planar, and furthermore, that it is -hard to test k-spine, 1-bend planarity. Given this complexity result, we further narrow our focus onto 2-spine, 1-bend planar drawings. We characterize 2-spine, 1-bend planarity using a new generalization of Hamiltonian graphs that we call Hamiltonian-with-handles graphs. We observe that our characterization naturally extends the connection between 2-page book embeddings and Hamiltonicity. Finally, we use our characterization to show that 2-outerplanar graphs are 2-spine, 1-bend planar. [Copyright &y& Elsevier]

Abstract: This paper studies the problem of computing orthogonal drawings of graphs with labels on vertices and edges. Our research is mainly motivated by Software Engineering and Information Systems domains, where tools like UML diagrams and ER-diagrams are considered fundamental for the design of sophisticated systems and/or complex data bases collecting enormous amount of information. A label is modeled as a rectangle of prescribed width and height and it can be associated with either a vertex or an edge. Our drawing algorithms guarantee no overlaps between labels, vertices, and edges and take advantage of the information about the set of labels to compute the geometry of the drawing. Several additional optimization goals are taken into account. Namely, the labeled drawing can be required to have either minimum total edge length, or minimum width, or minimum height, or minimum area among those preserving a given orthogonal representation. All these goals lead to NP-hard problems. We present MILP models to compute optimal drawings with respect to the first three goals and an exact algorithm that is based on these models to compute a labeled drawing of minimum area. We also present several heuristics for computing compact labeled orthogonal drawings and experimentally validate their performances, comparing their solutions against the optimum. [Copyright &y& Elsevier]

Let C be the family of 2D curves described by concave functions, let G be a planar graph, and let L be a linear ordering of the vertices of G. L is a curve embedding of G if for any given curve Λ∈C there exists a planar drawing of G such that: (i) the vertices are constrained to be on Λ with the same ordering as in L, and (ii) the edges are polylines with at most one bend. Informally speaking, a curve embedding can be regarded as a two-page book embedding in which the spine is bent. Although deciding whether a graph has a two-page book embedding is an NP-hard problem, in this paper it is proven that every planar graph has a curve embedding which can be computed in linear time. Applications of the concept of curve embedding to upward drawability and point-set embeddability problems are also presented. [Copyright &y& Elsevier]

The determination of IgG levels and their subclasses can provide clinically relevant information on the status of the immune system. Here we determined the sensitivity and reproducibility of the quantification of IgG subclasses from Dried Blood Spots (DBS) in Malawian uninfected infants exposed to HIV (HEU). Sixty paired samples of serum and DBS from HEU infants were used. Samples were collected from 1, 6, and 24-month old infants. IgGs concentrations from both serum and DBS were analyzed by BN ProSpec Siemens assay, using a different setting for sample dilutions. The reproducibility of the DBS method was tested on 10 samples run twice, starting from the DBS extraction process. To assess the systematic, proportional, and random differences, we computed the Passing-Bablok regression, and the Bland-Altman analysis to estimate the total mean bias between the two tests. The IgG isotypes concentrations from serum and DBS showed significant differences in all the comparisons. Generally, the DBS method underestimated IgG subclasses' values showing a recovery range between 51.2% and 77.6%. Passing Bablok regression on age-based groups showed agreement for IgG, IgG1, and IgG2, but not for IgG3 and IgG4. The mean bias obtained with the Bland Altman test varied largely depending on IgG isotypes (−0.02–2.21 g/l) Coefficient of variation <7.0% was found in the repeated tests for IgG, IgG1, IgG3, and IgG4, while it was 12.4% for IgG2. Varying degrees of differences were seen in the IgGs measurement in the two different matrices. In IgGs analysis, the DBS method offers promise for population-based research, but the results should be carefully evaluated and considered as a relative value since they are not equivalent to the serum concentrations. [ABSTRACT FROM AUTHOR]

Abstract A topological graph is a graph drawn in the plane. A topological graph is k -plane, k > 0 , if each edge is crossed at most k times. We study the problem of partitioning the edges of a k -plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for k = 1 , we focus on optimal 2-plane and on optimal 3-plane graphs, which are 2-plane and 3-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple (i.e., with neither self-loops nor parallel edges) optimal 2-plane graph into a 1-plane graph and a forest, while (ii) an edge partition formed by a 1-plane graph and two plane forests always exists and can be computed in linear time. (iii) There exist efficient algorithms to partition the edges of a simple optimal 2-plane graph into a 1-plane graph and a plane graph with maximum vertex degree at most 12, or with maximum vertex degree at most 8 if the optimal2-plane graph is such that its crossing-free edges form a graph with no separating triangles. (iv) There exists an infinite family of simple optimal 2-plane graphs such that in any edge partition composed of a 1-plane graph and a plane graph, the plane graph has maximum vertex degree at least 6 and the 1-plane graph has maximum vertex degree at least 12. (v) Every optimal 3-plane graph whose crossing-free edges form a biconnected graph can be decomposed, in linear time, into a 2-plane graph and two plane forests. [ABSTRACT FROM AUTHOR]

Visibility representations are a well-known paradigm to represent graphs. From a high-level perspective, a visibility representation of a graph G maps the vertices of G to non-overlapping geometric objects and the edges of G to visibilities, i.e., segments that do not intersect any geometric object other than at their end-points. In this paper, we initiate the study of 2.5D box visibility representations (2.5D-BRs) where vertices are mapped to 3D boxes having the bottom face in the plane z = 0 and edges are unobstructed lines of sight parallel to the x - or y -axis. We prove that: ( i ) Every complete bipartite graph admits a 2.5D-BR; ( i i ) The complete graph K n admits a 2.5D-BR if and only if n ⩽ 19 ; ( i i i ) Every graph with pathwidth at most 7 admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBRs), which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an n -vertex graph that admits a 2.5D-GBR has at most 4 n − 6 n edges and this bound is tight. Finally, we prove that deciding whether a given graph G admits a 2.5D-GBR with a given footprint is NP-complete (the footprint of a 2.5D-BR Γ is the set of bottom faces of the boxes in Γ). [ABSTRACT FROM AUTHOR]

A 1 -plane graph is a graph embedded in the plane such that each edge is crossed at most once. A NIC-plane graph is a 1-plane graph such that any two pairs of crossing edges share at most one end-vertex. An edge partition of a 1-plane graph G is a coloring of the edges of G with two colors, red and blue, such that both the graph induced by the red edges and the graph induced by the blue edges are plane graphs. We prove the following: ( i ) Every NIC-plane graph admits an edge partition such that the red graph has maximum vertex degree three; this bound on the vertex degree is worst-case optimal. ( ii ) Deciding whether a NIC-plane graph admits an edge partition such that the red graph has maximum vertex degree two is NP-complete. ( iii ) Deciding whether a 1-plane graph admits an edge partition such that the red graph has maximum vertex degree one, and computing one in the positive case, can be done in quadratic time. Applications of these results to graph drawing are also discussed. [ABSTRACT FROM AUTHOR]

Content-based travel recommender systems suggest touristic attractions based on a best match between users’ preferences and a given set of points of interests, called POIs for short. When designing such systems, a critical aspect is to equip them with a rich enough knowledge base that, for each POI, indicates how much the POI is relevant for a set of possible topics of interests, also called TOIs for short. This paper focuses on the problem of designing the Content Analyzer of a content-based travel recommender system. The Content Analyzer is a module that receives as input a set of POIs and a set of TOIs and it computes the relevance of each POI with respect to each TOI. The proposed approach is unsupervised, fully automatic, and it relies on publicly available sources of information. We describe an implementation of the technique in a system called Cicero and present an experimental evaluation of its effectiveness against a ground truth generated by experts. [ABSTRACT FROM AUTHOR]

We give new results about the relationship between 1-planar graphs and RAC graphs . A graph is 1-planar if it has a drawing where each edge is crossed at most once. A graph is RAC if it can be drawn in such a way that its edges cross only at right angles. These two classes of graphs and their relationships have been widely investigated in the last years, due to their relevance in application domains where computing readable graph layouts is important to analyze or design relational data sets. We study IC-planar graphs , the sub-family of 1-planar graphs that admit 1-planar drawings with independent crossings (i.e., no two crossed edges share an endpoint). We prove that every IC-planar graph admits a straight-line RAC drawing, which may require however exponential area. If we do not require right angle crossings, we can draw every IC-planar graph with straight-line edges in linear time and quadratic area. We then study the problem of testing whether a graph is IC-planar. We prove that this problem is NP-hard, even if a rotation system for the graph is fixed. On the positive side, we describe a polynomial-time algorithm that tests whether a triangulated plane graph augmented with a given set of edges that form a matching is IC-planar. [ABSTRACT FROM AUTHOR]

• Given the growing importance of the early identification of frailty to prevent subsequent functional decline and disability at both individual and population-level, we have conducted an international delphi study to identify defining characteristics of pre-frailty. • The final consensus statement produced by this international collaboration of experts in the field describes pre-frailty as an aged-associated, multi-factorial, multi-dimensional, and non-linear prodromal risk-state associated with one or more of physical impairment, cognitive decline, nutritional deficiencies, and socioeconomic inequalities, predisposing to the development of frailty. • Further study is required to fully operationalise this definition of pre-frailty for use in clinical practice including the development of improved screening and assessment tools. : Frailty is associated with a prodromal stage called pre-frailty, a potentially reversible and highly prevalent intermediate state before frailty becomes established. Despite being widely-used in the literature and increasingly in clinical practice, it is poorly understood. : To establish consensus on the construct and approaches to diagnose and manage pre-frailty. : We conducted a modified (electronic, two-round) Delphi consensus study. The questionnaire included statements concerning the concept, aspects and causes, types, mechanism, assessment, consequences, prevention and management of pre-frailty. Qualitative and quantitative analysis methods were employed. An agreement level of 70% was applied. : Twenty-three experts with different backgrounds from 12 countries participated. In total, 70 statements were circulated in Round 1. Of these, 52.8% were accepted. Following comments, 51 statements were re-circulated in Round 2 and 92.1% were accepted. It was agreed that physical and non-physical factors including psychological and social capacity are involved in the development of pre-frailty, potentially adversely affecting health and health-related quality of life. Experts considered pre-frailty to be an age-associated multi-factorial, multi-dimensional, and non-linear process that does not inevitably lead to frailty. It can be reversed or attenuated by targeted interventions. Brief, feasible, and validated tools and multidimensional assessment are recommended to identify pre-frailty. : Consensus suggests that pre-frailty lies along the frailty continuum. It is a multidimensional risk-state associated with one or more of physical impairment, cognitive decline, nutritional deficiencies and socioeconomic disadvantages, predisposing to the development of frailty. More research is needed to agree an operational definition and optimal management strategies. [ABSTRACT FROM AUTHOR]

Abstract: A 1-planar graph is a graph that can be embedded in the plane with at most one crossing per edge. It is known that testing 1-planarity of a graph is NP-complete. In this paper, we consider maximal 1-planar graphs. A graph G is maximal 1-planar if addition of any edge destroys 1-planarity of G. We first study combinatorial properties of maximal 1-planar embeddings. In particular, we show that in a maximal 1-planar embedding, the graph induced by the non-crossing edges is spanning and biconnected. Using the properties, we show that the problem of testing maximal 1-planarity of a graph G can be solved in linear time, if a rotation system Φ (i.e., the circular ordering of edges for each vertex) is given. We also prove that there is at most one maximal 1-planar embedding ξ of G that is consistent with the given rotation system Φ. Our algorithm also produces such an embedding in linear time, if it exists. [Copyright &y& Elsevier]

Abstract: We introduce and study a generalization of the well-known region of influence proximity drawings, called -proximity drawings. Intuitively, given a definition of proximity and two real numbers and , an -proximity drawing of a graph is a planar straight-line drawing Γ such that: (i) for every pair of adjacent vertices u, v, their proximity region “shrunk” by the multiplicative factor does not contain any vertices of Γ; (ii) for every pair of non-adjacent vertices u, v, their proximity region “expanded” by the factor contains some vertices of Γ other than u and v. In particular, the locations of the vertices in such a drawing do not always completely determine which edges must be present/absent, giving us some freedom of choice. We show that this generalization significantly enlarges the family of representable planar graphs for relevant definitions of proximity drawings, including Gabriel drawings, Delaunay drawings, and β-drawings, even for arbitrarily small values of and . We also study the extremal case of -proximity drawings, which generalize the well-known weak proximity drawing paradigm. [Copyright &y& Elsevier]

Abstract: The crossing resolution of a geometric graph is the minimum crossing angle at which any two edges cross each other. In this paper, we present upper and lower bounds to the crossing resolution of the complete geometric graphs. [Copyright &y& Elsevier]

Abstract: In this paper we study the problem of computing an upward straight-line embedding of a planar DAG (directed acyclic graph) G into a point set S, i.e. a planar drawing of G such that each vertex is mapped to a point of S, each edge is drawn as a straight-line segment, and all the edges are oriented according to a common direction. In particular, we show that no biconnected DAG admits an upward straight-line embedding into every point set in convex position. We provide a characterization of the family of DAGs that admit an upward straight-line embedding into every convex point set such that the points with the largest and the smallest y-coordinate are consecutive in the convex hull of the point set. We characterize the family of DAGs that contain a Hamiltonian directed path and that admit an upward straight-line embedding into every point set in general position. We also prove that a DAG whose underlying graph is a tree does not always have an upward straight-line embedding into a point set in convex position and we describe how to construct such an embedding for a DAG whose underlying graph is a path. Finally, we give results about the embeddability of some sub-classes of DAGs whose underlying graphs are trees on point set in convex and in general position. [Copyright &y& Elsevier]

In a visibility representation of a graph G , the vertices are represented by non-overlapping geometric objects, while the edges are represented as segments that only intersect the geometric objects associated with their end-vertices. Given a set P of n points, an Anchored Visibility Representation of a graph G with n vertices is a visibility representation such that for each vertex v of G , the geometric object representing v contains a point of P. We prove positive and negative results about the existence of anchored visibility representations under various models, both in 2D and in 3D space. We consider the case when the mapping between the vertices and the points is not given and the case when it is only partially given. [ABSTRACT FROM AUTHOR]