Bar 1-Visibility Drawings of 1-Planar Graphs

A bar 1-visibility drawing of a graph $G$ is a drawing of $G$ where each vertex is drawn as a horizontal line segment called a bar, each edge is drawn as a vertical line segment where the vertical line segment representing an edge must connect the horizontal line segments representing the end vertices and a vertical line segment corresponding to an edge intersects at most one bar which is not an end point of the edge. A graph $G$ is bar 1-visible if $G$ has a bar 1-visibility drawing. A graph $G$ is 1-planar if $G$ has a drawing in a 2-dimensional plane such that an edge crosses at most one other edge. In this paper we give linear-time algorithms to find bar 1-visibility drawings of diagonal grid graphs and maximal outer 1-planar graphs. We also show that recursive quadrangle 1-planar graphs and pseudo double wheel 1-planar graphs are bar 1-visible graphs.Comment: 15 pages, 9 figure

Prescription Narratives: Re-reading Joseph Cornell’s Pharmacy Series as Modernist Anti-dote

‘The Apothecary’s Chest: Magic, Art and Medication’ was a one-day symposium held at the University of Glasgow on November 24, 2007. The symposium called for a discussion on the evolution of the notions of mysticism, knowledge and superstition in the way they are intertwined in both science and the literary imagination in the figure of healers such as the apothecary, the alchemist, the shaman. There were three main areas of interest. The first involved traditional perceptions of physicians, who combined knowledge and superstition and thus bordered, in their practices, on the sphere of the occult. The second theme, evolving from the first, proposed an inquiry of the overlapping interests and processes of science, magic and prophesy, as well as of the implications and consequences of a privileged access to medical knowledge, while the third subject of discussion concentrated on the development of the symbolism of the healer in literature, history, philosophy of science, anthropology, theology, film and art. The twelve papers included in this volume, papers presented by doctoral candidates and young scholars from across a range of geographical regions and disciplines, result in a collection of approaches to an investigative field with topics ranging from mystical traits of mundane materials to the origins of the occult and gender struggles. The thirteenth and final essay included in the volume, Professor Bill Herbert’s ‘From Mere Bellies to the Bad Shaman’, is an exploration of the modern role of the contemporary poet in the form of an extended conversation initiated at the closing of the conference, when Professor Herbert was asked to combine a poetry reading with a few observations on the relationship between the poet and the shaman

On RAC Drawings of Graphs with One Bend per Edge

A k-bend right-angle-crossing drawing or (k-bend RAC drawing}, for short) of a graph is a polyline drawing where each edge has at most k bends and the angles formed at the crossing points of the edges are 90 degrees. Accordingly, a graph that admits a k-bend RAC drawing is referred to as k-bend right-angle-crossing graph (or k-bend RAC, for short). In this paper, we continue the study of the maximum edge-density of 1-bend RAC graphs. We show that an n-vertex 1-bend RAC graph cannot have more than $5.5n-O(1)$ edges. We also demonstrate that there exist infinitely many n-vertex 1-bend RAC graphs with exactly $5n-O(1)$ edges. Our results improve both the previously known best upper bound of $6.5n-O(1)$ edges and the corresponding lower bound of $4.5n-O(\sqrt{n})$ edges by Arikushi et al. (Comput. Geom. 45(4), 169--177 (2012)).Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

A Heuristic Approach Towards Drawings of Graphs with High Crossing Resolution

The crossing resolution of a non-planar drawing of a graph is the value of the minimum angle formed by any pair of crossing edges. Recent experiments have shown that the larger the crossing resolution is, the easier it is to read and interpret a drawing of a graph. However, maximizing the crossing resolution turns out to be an NP-hard problem in general and only heuristic algorithms are known that are mainly based on appropriately adjusting force-directed algorithms. In this paper, we propose a new heuristic algorithm for the crossing resolution maximization problem and we experimentally compare it against the known approaches from the literature. Our experimental evaluation indicates that the new heuristic produces drawings with better crossing resolution, but this comes at the cost of slightly higher aspect ratio, especially when the input graph is large.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

1-Bend RAC Drawings of 1-Planar Graphs

A graph is 1-planar if it has a drawing where each edge is crossed at most once. A drawing is RAC (Right Angle Crossing) if the edges cross only at right angles. The relationships between 1-planar graphs and RAC drawings have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC drawable and graphs that have a straight-line RAC drawing but that are not 1-planar [22]. Also, straight-line RAC drawings always exist for IC-planar graphs [9], a subclass of 1-planar graphs. One of the main questions still open is whether every 1-planar graph has a RAC drawing with at most one bend per edge. We positively answer this question