β\beta-Stars or On Extending a Drawing of a Connected Subgraph

We consider the problem of extending the drawing of a subgraph of a given plane graph to a drawing of the entire graph using straight-line and polyline edges. We define the notion of star complexity of a polygon and show that a drawing $\Gamma_H$ of an induced connected subgraph $H$ can be extended with at most $\min\{ h/2, \beta + \log_2(h) + 1\}$ bends per edge, where $\beta$ is the largest star complexity of a face of $\Gamma_H$ and $h$ is the size of the largest face of $H$. This result significantly improves the previously known upper bound of $72|V(H)|$ [5] for the case where $H$ is connected. We also show that our bound is worst case optimal up to a small additive constant. Additionally, we provide an indication of complexity of the problem of testing whether a star-shaped inner face can be extended to a straight-line drawing of the graph; this is in contrast to the fact that the same problem is solvable in linear time for the case of star-shaped outer face [9] and convex inner face [13].Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

Contact Representations of Graphs in 3D

We study contact representations of graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We show that for every 3-connected planar graph, there exists a simultaneous representation of the graph and its dual with 3D boxes. We give a linear-time algorithm for constructing such a representation. This result extends the existing primal-dual contact representations of planar graphs in 2D using circles and triangles. While contact graphs in 2D directly correspond to planar graphs, we next study representations of non-planar graphs in 3D. In particular we consider representations of optimal 1-planar graphs. A graph is 1-planar if there exists a drawing in the plane where each edge is crossed at most once, and an optimal n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a linear-time algorithm for representing optimal 1-planar graphs without separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph admits a representation with boxes. Hence, we consider contact representations with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a quadratic-time algorithm for representing optimal 1-planar graph with L-shaped polyhedra

Planar Drawings of Fixed-Mobile Bigraphs

A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In the most general case, we show NP-hardness. For k=0 and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of "layered" 1-bend drawings

Colored Point-Set Embeddings of Acyclic Graphs.

We show that any planar drawing of a forest of three stars whose vertices are constrained to be at fixed vertex locations may require $\Omega(n^\frac{2}{3})$ edges each having $\Omega(n^\frac{1}{3})$ bends in the worst case. The lower bound holds even when the function that maps vertices to points is not a bijection but it is defined by a 3-coloring. In contrast, a constant number of bends per edge can be obtained for 3-colored paths and for 3-colored caterpillars whose leaves all have the same color. Such results answer to a long standing open problem.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

The {Mondshein} Sequence

Canonical orderings [STOC'88, FOCS'92] have been used as a key tool in graph drawing, graph encoding and visibility representations for the last decades. We study a far-reaching generalization of canonical orderings to non-planar graphs that was published by Lee Mondshein in a PhD-thesis at M.I.T.\ as early as 1971. Mondshein proposed to order the vertices of a graph in a sequence such that, for any $i$, the vertices from $1$ to $i$ induce essentially a $2$-connected graph while the remaining vertices from $i+1$ to $n$ induce a connected graph. Mondshein's sequence generalizes canonical orderings and became later and independently known under the name \emph{non-separating ear decomposition}. Currently, the best known algorithm for computing this sequence achieves a running time of $O(nm)$; the main open problem in Mondshein's and follow-up work is to improve this running time to a subquadratic time. In this paper, we present the first algorithm that computes a Mondshein sequence in time and space $O(m)$, improving the previous best running time by a factor of $n$. In addition, we illustrate the impact of this result by deducing linear-time algorithms for several other problems, for which the previous best running times have been quadratic. In particular, we show how to compute three independent spanning trees in a $3$-connected graph in linear time, improving a result of Cheriyan and Maheshwari [J. Algorithms 9(4)]. Secondly, we improve the preprocessing time for the output-sensitive data structure by Di Battista, Tamassia and Vismara [Algorithmica 23(4)] that reports three internally disjoint paths between any given vertex pair from $O(n^2)$ to $O(m)$. Finally, we show how a very simple linear-time planarity test can be derived once a Mondshein sequence is computed

Einsatzmoeglichkeiten von Pflanzenoelen, insbesondere Rapsoel als Isoliermedium fuer elektrische Betriebsmittel der Energie- bzw. Hochspannungstechnik Abschlussbericht

Fuer den vorliegenden Bericht wurde die Eignung von Rapsoel als elektrischen Isolierstoff fuer den Einsatz in Betriebsmitteln zur Uebertragung und Verteilung elektrischer Energie, z. B. Transformatoren, Wandler und Kondensatoren, untersucht. Die prinzipielle Eignung von Rapsoel als Ersatz fuer handelsuebliche Isolierfluessigkeiten auf Mineraloelbasis laesst sich durch Untersuchung der elektrischen Eigenschaften erforschen. Ihre Praktikabilitaet zeigt eine abschliessende Pruefung eines rapsoelisolierten 20/0,4 kV Verteiltransformators. Die Ergebnisse zeigen, dass die Einflussfaktoren auf die elektrischen Kenngroessen eines Isolieroels auf Mineraloelbasis qualitativ fuer Rapsoel gelten. Fuer die Bewertung des Einflusses von Wasser, ist das deutlich hoehere Wasserloesungsvermoegen von Rapsoel zu beachten. Es liess sich nachweisen, dass Rapsoel ueber guenstige Voraussetzungen fuer den Einsatz in Betriebsmitteln der Energieversorgung, insbesondere Transformatoren, verfuegt. (orig.)Investigations of the electric properties of rapeseed oil provided information on its applicability as an insulating fluid in electric systems, which was verified with a rape-oil-insulated 20/0.4 kV distributing transformer. The findings suggest that the influencing parameters of mineral oil base insulating fluids apply also to rape-oil qualtitatively. The higher water solubility of rapeseed oil must be taken into account. It was proved that rapeseed oil is a favourable options in electric systems, especially transformers.Available from TIB Hannover: F03B575 / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEFachagentur Nachwachsende Rohstoffe e.V., Guelzow (Germany); Bundesministerium fuer Verbraucherschutz, Ernaehrung und Landwirtschaft (BMVEL), Berlin (Germany)DEGerman

Propagation and structure of streamers in liquid dielectrics

International audienceOur purpose is to present a critical review of the current understanding of streamer propagation in dielectric liquids in order to help define the direction of future research. We show that the molecular structure has a significant effect on streamer propagation. The main parameter affecting propagation is the electronic affinity of the liquid molecule

On Graphs Supported by Line Sets

For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they intersect at a common point, are more powerful than a set of parallel lines. In particular, we prove that every such set of lines supports outerpaths, lobsters, and squids, none of which are supported by any set of parallel lines. On the negative side, we prove that no set of n lines that intersect in a common point supports all n-vertex planar graphs. Finally, we show that there exists a set of n lines in general position that does not support all n-vertex planar graphs

Embeddability Problems for Upward Planar Digraphs

We study two embedding problems for upward planar digraphs. Both problems arise in the context of drawing sequences of upward planar digraphs having the same set of vertices, where the location of each vertex is to remain the same for all the drawings of the graphs. We develop a method, based on the notion of book embedding, that gives characterization results for embeddability as well as testing and drawing algorithms