Search

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]

• 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]

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]

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]