1,461 research outputs found

    Optimal Morphs of Convex Drawings

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    We give an algorithm to compute a morph between any two convex drawings of the same plane graph. The morph preserves the convexity of the drawing at any time instant and moves each vertex along a piecewise linear curve with linear complexity. The linear bound is asymptotically optimal in the worst case.Comment: To appear in SoCG 201

    Convexity-Increasing Morphs of Planar Graphs

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    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

    Morphing Planar Graph Drawings Optimally

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    We provide an algorithm for computing a planar morph between any two planar straight-line drawings of any nn-vertex plane graph in O(n)O(n) morphing steps, thus improving upon the previously best known O(n2)O(n^2) upper bound. Further, we prove that our algorithm is optimal, that is, we show that there exist two planar straight-line drawings Γs\Gamma_s and Γt\Gamma_t of an nn-vertex plane graph GG such that any planar morph between Γs\Gamma_s and Γt\Gamma_t requires Ω(n)\Omega(n) morphing steps

    Pole Dancing: 3D Morphs for Tree Drawings

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    We study the question whether a crossing-free 3D morph between two straight-line drawings of an nn-vertex tree can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with O(log⁥n)O(\log n) steps, while for the latter Θ(n)\Theta(n) steps are always sufficient and sometimes necessary.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

    On contractible edges in convex decompositions

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    Let Π\Pi be a convex decomposition of a set PP of n≄3n\geq 3 points in general position in the plane. If Π\Pi consists of more than one polygon, then either Π\Pi contains a deletable edge or Π\Pi contains a contractible edge

    A Note on Plus-Contacts, Rectangular Duals, and Box-Orthogonal Drawings

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    A plus-contact representation of a planar graph GG is called cc-balanced if for every plus shape +v+_v, the number of other plus shapes incident to each arm of +v+_v is at most cΔ+O(1) c \Delta +O(1), where Δ\Delta is the maximum degree of GG. Although small values of cc have been achieved for a few subclasses of planar graphs (e.g., 22- and 33-trees), it is unknown whether cc-balanced representations with c<1c<1 exist for arbitrary planar graphs. In this paper we compute (1/2)(1/2)-balanced plus-contact representations for all planar graphs that admit a rectangular dual. Our result implies that any graph with a rectangular dual has a 1-bend box-orthogonal drawings such that for each vertex vv, the box representing vv is a square of side length deg(v)2+O(1)\frac{deg(v)}{2}+ O(1).Comment: A poster related to this research appeared at the 25th International Symposium on Graph Drawing & Network Visualization (GD 2017
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