504,951 research outputs found

    Transforming planar graph drawings while maintaining height

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    There are numerous styles of planar graph drawings, notably straight-line drawings, poly-line drawings, orthogonal graph drawings and visibility representations. In this note, we show that many of these drawings can be transformed from one style to another without changing the height of the drawing. We then give some applications of these transformations

    Convex drawings of the complete graph: topology meets geometry

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    In this work, we introduce and develop a theory of convex drawings of the complete graph KnK_n in the sphere. A drawing DD of KnK_n is convex if, for every 3-cycle TT of KnK_n, there is a closed disc ΔT\Delta_T bounded by D[T]D[T] such that, for any two vertices u,vu,v with D[u]D[u] and D[v]D[v] both in ΔT\Delta_T, the entire edge D[uv]D[uv] is also contained in ΔT\Delta_T. As one application of this perspective, we consider drawings containing a non-convex K5K_5 that has restrictions on its extensions to drawings of K7K_7. For each such drawing, we use convexity to produce a new drawing with fewer crossings. This is the first example of local considerations providing sufficient conditions for suboptimality. In particular, we do not compare the number of crossings {with the number of crossings in} any known drawings. This result sheds light on Aichholzer's computer proof (personal communication) showing that, for n12n\le 12, every optimal drawing of KnK_n is convex. Convex drawings are characterized by excluding two of the five drawings of K5K_5. Two refinements of convex drawings are h-convex and f-convex drawings. The latter have been shown by Aichholzer et al (Deciding monotonicity of good drawings of the complete graph, Proc.~XVI Spanish Meeting on Computational Geometry (EGC 2015), 2015) and, independently, the authors of the current article (Levi's Lemma, pseudolinear drawings of KnK_n, and empty triangles, \rbr{J. Graph Theory DOI: 10.1002/jgt.22167)}, to be equivalent to pseudolinear drawings. Also, h-convex drawings are equivalent to pseudospherical drawings as demonstrated recently by Arroyo et al (Extending drawings of complete graphs into arrangements of pseudocircles, submitted)

    That Some of Sol Lewitt's Later Wall Drawings Aren't Wall Drawings

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    Sol LeWitt is probably most famous for wall drawings. They are an extension of work he had done in sculpture and on paper, in which a simple rule specifies permutations and variations of elements. With wall drawings, the rule is given for marks to be made on a wall. We should distinguish these algorithmic works from impossible-to-implement instruction works and works realized by following preparatory sketches. Taking the core feature of a wall drawing to be that it is algorithmic, some of LeWitt's later works are wall drawings in name only

    On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings

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    We study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of max-degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal drawings with at most two segments per edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

    That Some of Sol Lewitt's Later Wall Drawings Aren't Wall Drawings

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    Sol LeWitt is probably most famous for wall drawings. They are an extension of work he had done in sculpture and on paper, in which a simple rule specifies permutations and variations of elements. With wall drawings, the rule is given for marks to be made on a wall. We should distinguish these algorithmic works from impossible-to-implement instruction works and works realized by following preparatory sketches. Taking the core feature of a wall drawing to be that it is algorithmic, some of LeWitt's later works are wall drawings in name only

    Strongly Monotone Drawings of Planar Graphs

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    A straight-line drawing of a graph is a monotone drawing if for each pair of vertices there is a path which is monotonically increasing in some direction, and it is called a strongly monotone drawing if the direction of monotonicity is given by the direction of the line segment connecting the two vertices. We present algorithms to compute crossing-free strongly monotone drawings for some classes of planar graphs; namely, 3-connected planar graphs, outerplanar graphs, and 2-trees. The drawings of 3-connected planar graphs are based on primal-dual circle packings. Our drawings of outerplanar graphs are based on a new algorithm that constructs strongly monotone drawings of trees which are also convex. For irreducible trees, these drawings are strictly convex
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