EPG-representations with small grid-size

In an EPG-representation of a graph $G$ each vertex is represented by a path in the rectangular grid, and $(v,w)$ is an edge in $G$ if and only if the paths representing $v$ an $w$ share a grid-edge. Requiring paths representing edges to be x-monotone or, even stronger, both x- and y-monotone gives rise to three natural variants of EPG-representations, one where edges have no monotonicity requirements and two with the aforementioned monotonicity requirements. The focus of this paper is understanding how small a grid can be achieved for such EPG-representations with respect to various graph parameters. We show that there are $m$-edge graphs that require a grid of area $\Omega(m)$ in any variant of EPG-representations. Similarly there are pathwidth-$k$ graphs that require height $\Omega(k)$ and area $\Omega(kn)$ in any variant of EPG-representations. We prove a matching upper bound of $O(kn)$ area for all pathwidth-$k$ graphs in the strongest model, the one where edges are required to be both x- and y-monotone. Thus in this strongest model, the result implies, for example, $O(n)$, $O(n \log n)$ and $O(n^{3/2})$ area bounds for bounded pathwidth graphs, bounded treewidth graphs and all classes of graphs that exclude a fixed minor, respectively. For the model with no restrictions on the monotonicity of the edges, stronger results can be achieved for some graph classes, for example an $O(n)$ area bound for bounded treewidth graphs and $O(n \log^2 n)$ bound for graphs of bounded genus.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Unsplittable coverings in the plane

A system of sets forms an {\em $m$-fold covering} of a set $X$ if every point of $X$ belongs to at least $m$ of its members. A $1$-fold covering is called a {\em covering}. The problem of splitting multiple coverings into several coverings was motivated by classical density estimates for {\em sphere packings} as well as by the {\em planar sensor cover problem}. It has been the prevailing conjecture for 35 years (settled in many special cases) that for every plane convex body $C$, there exists a constant $m=m(C)$ such that every $m$-fold covering of the plane with translates of $C$ splits into $2$ coverings. In the present paper, it is proved that this conjecture is false for the unit disk. The proof can be generalized to construct, for every $m$, an unsplittable $m$-fold covering of the plane with translates of any open convex body $C$ which has a smooth boundary with everywhere {\em positive curvature}. Somewhat surprisingly, {\em unbounded} open convex sets $C$ do not misbehave, they satisfy the conjecture: every $3$-fold covering of any region of the plane by translates of such a set $C$ splits into two coverings. To establish this result, we prove a general coloring theorem for hypergraphs of a special type: {\em shift-chains}. We also show that there is a constant $c>0$ such that, for any positive integer $m$, every $m$-fold covering of a region with unit disks splits into two coverings, provided that every point is covered by {\em at most} $c2^{m/2}$ sets

Intersection Graphs of L-Shapes and Segments in the Plane

An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: ⌊,⌈,⌋ and ⌉. A k-bend path is a simple path in the plane, whose direction changes k times from horizontal to vertical. If a graph admits an intersection representation in which every vertex is represented by an ⌊, an ⌊ or ⌈, a k-bend path, or a segment, then this graph is called an ⌊-graph, ⌊,⌈-graph, B k -VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer [Discrete Mathematics, 108(1):365–372, 1992], stating that every ⌊,⌈-graph is a SEG-graph, we investigate several known subclasses of SEG-graphs and show that they are ⌊-graphs, or B k -VPG-graphs for some small constant k. We show that all planar 3-trees, all line graphs of planar graphs, and all full subdivisions of planar graphs are ⌊-graphs. Furthermore we show that all complements of planar graphs are B 19-VPG-graphs and all complements of full subdivisions are B 2-VPG-graphs. Here a full subdivision is a graph in which each edge is subdivided at least once

Trabajadoras domésticas: los medios de comunicación y la identidad

v. 14, n.1, p.99-108, jan./jun. 2011.Submitted by Marlene Santos ([email protected]) on 2013-09-13T20:33:03Z No. of bitstreams: 1 15685-66172-1-PB.pdf: 202615 bytes, checksum: 8b3c444dba87f27a055e00b2ab1e8b02 (MD5)Approved for entry into archive by Claudia Moura ([email protected]) on 2013-10-13T19:10:08Z (GMT) No. of bitstreams: 1 15685-66172-1-PB.pdf: 202615 bytes, checksum: 8b3c444dba87f27a055e00b2ab1e8b02 (MD5)Approved for entry into archive by Claudia Moura ([email protected]) on 2013-10-13T19:15:07Z (GMT) No. of bitstreams: 1 15685-66172-1-PB.pdf: 202615 bytes, checksum: 8b3c444dba87f27a055e00b2ab1e8b02 (MD5)Made available in DSpace on 2013-10-13T19:15:23Z (GMT). No. of bitstreams: 1 15685-66172-1-PB.pdf: 202615 bytes, checksum: 8b3c444dba87f27a055e00b2ab1e8b02 (MD5) Previous issue date: 2011-06Programa de Apoio às Publicações Periódicas Científicas (PROAPUPEC) da UFGEste artigo traz dados e reflexões feitos a partir de uma pesquisa em que se trabalha a relação de 31 trabalhadoras domésticas de Goiânia, com mídia, beleza e consumo. Claro que, ao se mergulhar em histórias de vida, os temas não surgem compartimentados e as falas acabam trazendo nuances de realidades que por vezes ultrapassam o objeto específico do estudo. Este artigo, portanto, traz estes feixes por onde se pôde observar as percepções de trabalhadoras domésticas sobre o seu trabalho, percorrendo linhas que ligam a ocupação, a partir de sua história, aos sentidos que estão entranhados na cultura hegemônica, passando por um dos grandes mediadores culturais da pós-modernidade, que é a mídia. Os meios de comunicação de massa exibem uma certa representação de trabalhadoras domésticas, que chama a atenção por suas semelhanças, o que sugere sentidos legitimados e naturalizados pela cultura. Isso pode repercutir no imaginário da sociedade sobre o que seja a ocupação, e, claro, incluindo as próprias trabalhadoras domésticas sobre o que pensam de si e do seu trabalho.This article presents data and reflections based on a research in which the relation of 31 domestic workers from Goiânia, with media, beauty and consumption has been worked out. Surely that when one plunges into life stories the themes do not appear compartmentalized and the lines end up bringing nuances of realities that sometimes go beyond the specific object of study. This article, therefore, brings these bundles through which it could be possible to observe the domestic workers perceptions about their work, going through the lines that conect the profession, from its history, to the meanings that are embedded in the hegemonic culture, passing by one of the greatest cultural mediators of post-modernity, that is the media. The mass media exhibit a certain representation of domestic workers, that calls attention to their resemblances, which suggests legitimized senses naturalized by the culture. This may reflect on the imaginary of society about what the profession would be, and, sure, includes domestic workers own thoughts about themselves and their work.Este artículo presenta los datos y reflexiones a partir de una investigación en la que se trabaja la relación de 31 trabajadoras domésticas de Goiânia, los medios de comunicación, la belleza y el consumo. Por supuesto, al profundizar en las historias de la vida, los temas no aparecen fragmentados y las líneas terminan llevando matices de las cosas que a veces van más allá del objeto específico de estudio. Este artículo, por lo tanto, portea estos paquetes en que se puede observar la visión de los trabajadores domésticos con referencia a su trabajo, que cubre las rutas desde la profesión, su historia hasta los significados que están incrustados en la cultura hegemónica, en busca de un gran mediador cultural de la posmodernidad, que es los medios de comunicación. Los medios de comunicación presentan una certa representación de los trabajadores domésticos, que llama la atención sobre sus similitudes, lo que sugiere formas legitimadas y naturalizadas por la cultura. Esto puede reflejar la imaginación de la sociedad en el que la ocupación y, por supuesto, incluí a las trabajadoras domésticas y lo que ellas piensan de sí mismas y de su trabajo

Monotone Simultaneous Embeddings of Paths in d Dimensions

We study the following problem: Given k paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that, for any dimension d, there is a set of d+1 paths that does not admit a monotone simultaneous geometric embedding