Introduction. A pseudoline is formed from a line by stretching the plane without tearing: it is the image of a line under a homeomorphism of the plane [13]. In arrangements of pseudolines, pairs of pseudolines intersect at most once and cross at their intersections. Pseudoline arrangements can be used to model sorting networks [1], tilings of convex polygons by rhombi [4], and graphs that have distance-preserving embeddings into hypercubes [6]. They are also closely related to oriented matroids [11]. We consider here the visualization of arrangements using well-shaped curves. Primarily, we study weak outerplanar pseudoline arrangements. An arrangement is weak if it does not necessarily have a crossing for every pair of pseudolines [12], and outerplanar if every crossing is part of an unbounded face of the arrangement. We show that these arrangements can be drawn with all curves convex, either as polygonal chains with at most two bends per pseudoline or as semicircles above a line. Arbitrary pseudolines can also be drawn as convex curves, but may require linearly many bends. Related Work. Several results related to the visualization of pseudoline arrangements are known. In wiring diagrams, pseudolines are drawn on parallel horizontal lines, with crossings on short line segments that connect pairs of horizontal lines [10]. The graphs of arrangements have drawings in small grids [8] and the dual graphs of weak arrangements have drawings in which each bounded face is centrally symmetric [5]. The pseudoline arrangements in which each pseudoline is a translated quadrant can be used to visualize learning spaces representing the states of a human learner [7]. Researchers in graph drawing have also studied force-directed methods for schematizing systems of curves representing metro maps by replacing each curve by a spline; these curves are not necessarily pseudolines, but they typically have few crossings [9]. Results. Below we state our results for outerplanar and arbitrary arrangements. Theorem 1. Every weak outerplanar pseudoline arrangement may be represented by
Several results related to the visualization of pseudoline arrangements are known. In , pseudolines are drawn on parallel horizontal lines, with crossings on short line segments that connect pairs of horizontal lines @cite_14 . Using a method based on compaction of wiring diagrams, the graphs of pseudoline arrangements may be given straight line drawings in small grids @cite_0 . The planar dual graph of a weak pseudoline arrangement may be characterized as having drawings in which each bounded face is a centrally symmetric polygon @cite_9 . The pseudoline arrangements in which each pseudoline is a translated quadrant can be used to visualize , representing the possible states of knowledge of a human learner @cite_8 . Researchers in graph drawing have also studied force-directed methods for schematizing systems of curves representing metro maps by replacing each curve by a spline; these curves are not necessarily pseudolines, but they typically have few crossings @cite_3 .
@cite_14 @cite_8 @cite_9 @cite_3 @cite_0
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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 On 7 6. No known input causes our algorithm to use area Ωn 1+e for any e>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. Given a planar triangulated graph (PTG) G, the problem of constructing a floor-plan F such that G is the dual of F and the boundary of F is rectangular is studied. It is shown that if only zero-concave rectilinear modules (CRM) (or rectangular modules) and 1-CRM (i.e., L-shaped) are allowed, there are PTGs that do not admit any floor-plan. However, if 2-bend modules (e.g., T-shaped and Z-shaped) are also allowed, then every biconnected PTG admits a floor-plan. Thus, the employment of 2-bend modules is necessary and sufficient for graph dualization floor-planning. A linear-time algorithm for constructing a 2-CRM floor-plan of an arbitrary PTG is proposed. This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two bipolar orientations that are transversal. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straight-line drawing algorithm for irreducible triangulations. For a random irreducible triangulation with n vertices, the grid size of the drawing is asymptotically with high probability 11n 27x11n 27 up to an additive error of O(n). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size (@?n 2@?-1)x@?n 2@?. Right Angle Crossing (RAC) drawings are polyline drawings where each crossing forms four right angles. RAC drawings have been introduced because cognitive experiments provided evidence that increasing the number of crossings does not decrease the readability of the drawing if the edges cross at right angles. We investigate to what extent RAC drawings can help in overcoming the limitations of widely adopted planar graph drawing conventions, providing both positive and negative results. First, we prove that there exist acyclic planar digraphs not admitting any straight-line upward RAC drawing and that the corresponding decision problem is NP-hard. Also, we show digraphs whose straight-line upward RAC drawings require exponential area. Second, we study if RAC drawings allow us to draw bounded-degree graphs with lower curve complexity than the one required by more constrained drawing conventions. We prove that every graph with vertex-degree at most 6 (at most 3) admits a RAC drawing with curve complexity 2 (resp. 1) and with quadratic area. Third, we consider a natural non-planar generalization of planar embedded graphs. Here we give bounds for curve complexity and area different from the ones known for planar embeddings.