L-Drawings of Directed Graphs

We introduce L-drawings, a novel paradigm for representing directed graphs aiming at combining the readability features of orthogonal drawings with the expressive power of matrix representations. In an L-drawing, vertices have exclusive $x$- and $y$-coordinates and edges consist of two segments, one exiting the source vertically and one entering the destination horizontally. We study the problem of computing L-drawings using minimum ink. We prove its NP-completeness and provide a heuristics based on a polynomial-time algorithm that adds a vertex to a drawing using the minimum additional ink. We performed an experimental analysis of the heuristics which confirms its effectiveness.Comment: 11 pages, 7 figure

Pixel and Voxel Representations of Graphs

We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for $k$-outerplanar graphs with $n$ vertices, $\Theta(kn)$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, $\Theta(n^2)$ voxels are always sufficient and sometimes necessary for any $n$-vertex graph. We improve this bound to $\Theta(n\cdot \tau)$ for graphs of treewidth $\tau$ and to $O((g+1)^2n\log^2n)$ for graphs of genus $g$. In particular, planar graphs admit representations with $O(n\log^2n)$ voxels

Improved Parameterized Algorithms for the Kemeny Aggregation Problem

We give improvements over fixed parameter tractable (FPT) algo-rithms to solve the Kemeny aggregation problem, where the task is to summarize a multi-set of preference lists, called votes, over a set of alternatives, called candidates, into a single preference list that has the minimum total τ-distance from the votes. The τ-distance between two preference lists is the number of pairs of candidates that are or-dered differently in the two lists. We study the problem for preference lists that are total orders. We develop algorithms of running times O∗(1.403kt), O∗(5.823kt/m) ≤ O∗(5.823kavg) and O∗(4.829kmax) for the problem, ignoring the polynomial factors in the O ∗ notation, where kt is the optimum total τ-distance, m is the number of votes, and kavg (resp, kmax) is the average (resp, maximum) over pairwise τ-distances of votes. Our algorithms improve the best previously known running times of O∗(1.53kt) and O∗(16kavg) ≤ O∗(16kmax) [4, 5], which also implies an O∗(164kt/m) running time. We also show how to enumerate all optimal solutions in O∗(36kt/m) ≤ O∗(36kavg) time.

Lombardi Drawings of Graphs

We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure

Ciliopathy is differentially distributed in the brain of a Bardet-Biedl syndrome mouse model

Bardet-Biedl syndrome (BBS) is a genetically heterogeneous inherited human disorder displaying a pleotropic phenotype. Many of the symptoms characterized in the human disease have been reproduced in animal models carrying deletions or knock-in mutations of genes causal for the disorder. Thinning of the cerebral cortex, enlargement of the lateral and third ventricles, and structural changes in cilia are among the pathologies documented in these animal models. Ciliopathy is of particular interest in light of recent studies that have implicated primary neuronal cilia (PNC) in neuronal signal transduction. In the present investigation, we tested the hypothesis that areas of the brain responsible for learning and memory formation would differentially exhibit PNC abnormalities in animals carrying a deletion of the Bbs4 gene (Bbs4-/-). Immunohistochemical localization of adenylyl cyclase-III (ACIII), a marker restricted to PNC, revealed dramatic alterations in PNC morphology and a statistically significant reduction in number of immunopositive cilia in the hippocampus and amygdala of Bbs4-/- mice compared to wild type (WT) littermates. Western blot analysis confirmed the decrease of ACIII levels in the hippocampus and amygdala of Bbs4-/- mice, and electron microscopy demonstrated pathological alterations of PNC in the hippocampus and amygdala. Importantly, no neuronal loss was found within the subregions of amygdala and hippocampus sampled in Bbs4-/- mice and there were no statistically significant alterations of ACIII immunopositive cilia in other areas of the brain not known to contribute to the BBS phenotype. Considered with data documenting a role of cilia in signal transduction these findings support the conclusion that alterations in cilia structure or neurochemical phenotypes may contribute to the cognitive deficits observed in the Bbs4-/- mouse mode. © 2014 Agassandian et al

Mutation analysis in Bardet-Biedl syndrome by DNA pooling and massively parallel resequencing in 105 individuals

Bardet–Biedl syndrome (BBS) is a rare, primarily autosomal-recessive ciliopathy. The phenotype of this pleiotropic disease includes retinitis pigmentosa, postaxial polydactyly, truncal obesity, learning disabilities, hypogonadism and renal anomalies, among others. To date, mutations in 15 genes (BBS1–BBS14, SDCCAG8) have been described to cause BBS. The broad genetic locus heterogeneity renders mutation screening time-consuming and expensive. We applied a strategy of DNA pooling and subsequent massively parallel resequencing (MPR) to screen individuals affected with BBS from 105 families for mutations in 12 known BBS genes. DNA was pooled in 5 pools of 21 individuals each. All 132 coding exons of BBS1–BBS12 were amplified by conventional PCR. Subsequent MPR was performed on an Illumina Genome Analyzer II(™) platform. Following mutation identification, the mutation carrier was assigned by CEL I endonuclease heteroduplex screening and confirmed by Sanger sequencing. In 29 out of 105 individuals (28%), both mutated alleles were identified in 10 different BBS genes. A total of 35 different disease-causing mutations were confirmed, of which 18 mutations were novel. In 12 additional families, a total of 12 different single heterozygous changes of uncertain pathogenicity were found. Thus, DNA pooling combined with MPR offers a valuable strategy for mutation analysis of large patient cohorts, especially in genetically heterogeneous diseases such as BBS

Orthogonal drawings with few layers

In this paper, we study 3-dimensional orthogonal graph drawings. Motivated by the fact that only a limited number of layers is possible in VLSI technology, and also noting that a small number of layers is easier to parse for humans, we study drawings where one dimension is restricted to be very small. We give algorithms to obtain point-drawings with 3layers and 4 bends per edge, and algorithms to obtain box-drawings with 2 layers and 2 bends per edge. Several other related results are included as well. Our constructions have optimal volume, which we prove by providing lower bounds

Drawing Planar Graphs with Reduced Height

A straight-line (respectively, polyline) drawing Γ of a planar graph G on a set L k of k parallel lines is a planar drawing that maps each vertex of G to a distinct point on L k and each edge of G to a straight line segment (respectively, a polygonal chain with the bends on L k ) between its endpoints. The height of Γ is k, i.e., the number of lines used in the drawing. In this paper we compute new upper bounds on the height of polyline drawings of planar graphs using planar separators. Specifically, we show that every n-vertex planar graph with maximum degree Δ, having a simple cycle separator of size λ, admits a polyline drawing with height 4n/9 + O(λΔ), where the previously best known bound was 2n/3. Since λ∈O(n√) , this implies the existence of a drawing of height at most 4n/9 + o(n) for any planar triangulation with Δ∈o(n√) . For n-vertex planar 3-trees, we compute straight-line drawings with height 4n/9 + O(1), which improves the previously best known upper bound of n/2. All these results can be viewed as an initial step towards compact drawings of planar triangulations via choosing a suitable embedding of the input graph