Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time

Thomassen characterized some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph is drawable in straight-lines if and only if it does not contain the configuration [C. Thomassen, Rectilinear drawings of graphs, J. Graph Theory, 10(3), 335-341, 1988]. In this paper, we characterize some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph can be re-embedded into a straight-line drawable 1-plane embedding of the same graph if and only if it does not contain the configuration. Re-embedding of a 1-plane embedding preserves the same set of pairs of crossing edges. We give a linear-time algorithm for finding a straight-line drawable 1-plane re-embedding or the forbidden configuration.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016). This is an extended abstract. For a full version of this paper, see Hong S-H, Nagamochi H.: Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time, Technical Report TR 2016-002, Department of Applied Mathematics and Physics, Kyoto University (2016

Convexity-Increasing Morphs of Planar Graphs

We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal. To obtain our result, we use a well-known technique by Hong and Nagamochi for finding redrawings with convex faces while preserving y-coordinates. Using a variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and Nagamochi's result which comes with a better running time. This is of independent interest, as Hong and Nagamochi's technique serves as a building block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201

Finding lines – A celebration of drawing and mark making.

A series of 10 drawings and one video (titled Faint/Feint) that explore process, performance and gesture, selected for the group exhibition ‘Finding Lines – A Celebration Of Drawing And Mark Making’ at Derby Museum and Art Gallery. The ten drawings for Finding Lines are not drawings, they are carbon copies made with small sheets of typewriter carbon paper placed underneath the paper that will be drawn on, and on top of a second sheet of paper which receives the impression of the drawing. Each drawing is made of a series of straight lines drawn with the aid of a set square. Faint/Feint privileges the most basic elements of drawing; pencil, line, paper and tool. The carbon copy is an ‘automatic’ record of the corporeal (and cognitive) act of drawing: it captures all the mistakes I make; the slips, smudges, misalignment and movement - and replicates them. The drawing is a poor performance of an activity that could easily be automated. I have approached drawing as a corporeal exercise that relies on concentration and stamina and which is always imperfect because in doing it I can never match the precision of the computer (although the carbon copy nods to the perfect copying of the photocopier and the printer). Faint/Feint 10 x A1 carbon copy drawings, 60gsm newsprint.N/

From on-line sketching to 2D and 3D geometry: A fuzzy knowledge based system

The paper describes the development of a fuzzy knowledge based prototype system for conceptual design. This real time system is designed to infer user’s sketching intentions, to segment sketched input and generate corresponding geometric primitives: straight lines, circles, arcs, ellipses, elliptical arcs, and B-spline curves. Topology information (connectivity, unitary constraints and pairwise constraints) is received dynamically from 2D sketched input and primitives. From the 2D topology information, a more accurate 2D geometry can be built up by applying a 2D geometric constraint solver. Subsequently, 3D geometry can be received feature by feature incrementally. Each feature can be recognised by inference knowledge in terms of matching its 2D primitive configurations and connection relationships. The system accepts not only sketched input, working as an automatic design tools, but also accepts user’s interactive input of both 2D primitives and special positional 3D primitives. This makes it easy and friendly to use. The system has been tested with a number of sketched inputs of 2D and 3D geometry

Contradictions in social enterprise: do they draw in straight lines or circles?

This paper provides a critical perspective on the discourse surrounding the concept of social enterprise. The paper shifts the lens away from numbers to consider how actors see themselves as social enterprises. The authors make sense of the foundations upon which the concept of social enterprise and entrepreneurship is ‘drawn’ – quite literally – by considering linear models and diagrams that analyse social enterprise on a continuum between non-profit (mission) and profit (market) orientation. A great deal has been made of the success and growth of social enterprise. The imagery in the literature reflects an emphasis on growth resulting from ‘the rising tide of commercialisation of non-profit organisations’ (Dees, 1998) with the result that the CBI now includes over 50,000 organisations in a social enterprise sector (SBS, 2005). Despite reports of rapid growth, there is awareness that ‘take-up of social enterprise model … is patchy and fails to reflect the enthusiasm with which it is discussed’ (Stevenson in Westall &amp; Chalkley 2007). We ask why? A methodological approach involving visual drawings by actors reveals stories and sensemaking experiences of social enterprises. Open conversations enabled the researchers to gain deep insights that would not have been as insightful through a quantitative approach. The key findings suggest: Firstly, participants report tensions when pursuing social and economic goals simultaneously. Secondly, whilst some welcome opportunities that are emerging, others perceive substantive threats to the third sector. Thirdly, Social enterprise emerges as a diverse and heterogeneous movement located at the boundaries of public, private and voluntary sectors. At each boundary, different constitutional forms and practices are seen. In conclusion, it is argued that the linear perspective itself gives the impression that there is a ‘patchy’ take up of social enterprise. A heterogeneous perspective reveals that theory and policy development is patchy, rather than social enterprise practices. The unique contribution this research paper offers is within the depth of enquiry and insight into the actual practices provided from those within the field. The critical perspective is taken from the literature and discussed in the settings of the actors in the field which provides practitioners, business support agencies and academics with a different level of empirical investigation that captures an originality and narrative that has barely been explored before.</p

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

Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing, Bordeaux, 201

A Coloring Algorithm for Disambiguating Graph and Map Drawings

Drawings of non-planar graphs always result in edge crossings. When there are many edges crossing at small angles, it is often difficult to follow these edges, because of the multiple visual paths resulted from the crossings that slow down eye movements. In this paper we propose an algorithm that disambiguates the edges with automatic selection of distinctive colors. Our proposed algorithm computes a near optimal color assignment of a dual collision graph, using a novel branch-and-bound procedure applied to a space decomposition of the color gamut. We give examples demonstrating the effectiveness of this approach in clarifying drawings of real world graphs and maps

Visuality and the haptic qualities of the line in generative art

The line has an important and particular relationship with the generative artwork distinct from other elements such as the ‘pixel’, ‘voxel’ or the ‘points’ that make up point clouds. The line has a dual nature as both continuous and discrete which makes it perhaps uniquely placed to straddle the analog and digital worlds. It has a haptic or felt quality as well as an inherent ambiguity that promotes a relatively active interpretive role for the audience. There is an extensive history of the line in generative systems and artworks, taking both analog and digital forms. That it continues to play an important role, alongside other more photographically inspired ‘perceptual schemas’, may be a testament to its enduring usefulness and unique character. This paper considers the particular affordances and the ‘visuality’ of the line in relation to generative artworks. This includes asking how we might account for the felt quality of lines and the socially and culturally constructed aspects that shape our relationship with them. It asks whether, in what has been described as a ‘post digital’ or even ‘post post digital’ world, the line may offer a way to re-emphasise a more human scale and a materiality that can push back, gently, against other more dominant perceptual schemas. It also asks what generative art can learn from drawing theory, many of the concerns of which parallel and intersect with those of generative art