7,072 research outputs found

    Planar Octilinear Drawings with One Bend Per Edge

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    In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal (4545^\circ) line-segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A kk-planar graph is a planar graph in which each vertex has degree less or equal to kk. In particular, we prove that every 4-planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size O(n2)×O(n)O(n^2) \times O(n). For 5-planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in super-polynomial area. However, for 6-planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge

    Outerplanar graph drawings with few slopes

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    We consider straight-line outerplanar drawings of outerplanar graphs in which a small number of distinct edge slopes are used, that is, the segments representing edges are parallel to a small number of directions. We prove that Δ1\Delta-1 edge slopes suffice for every outerplanar graph with maximum degree Δ4\Delta\ge 4. This improves on the previous bound of O(Δ5)O(\Delta^5), which was shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound is tight: for every Δ4\Delta\ge 4 there is an outerplanar graph with maximum degree Δ\Delta that requires at least Δ1\Delta-1 distinct edge slopes in an outerplanar straight-line drawing.Comment: Major revision of the whole pape

    Optimal partisan districting on planar geographies

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    We show that optimal partisan districting in the plane with geographical constraints is an NP-complete problem

    Decomposition of Geometric Set Systems and Graphs

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    We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opposite in 3 dimensions, for any polyhedron and any mm we construct an m-fold covering of the space that is not decomposable. We also give constructions that show that concave polygons are usually not cover-decomposable. We start the first part with a detailed survey of all results on the cover-decomposability of polygons. The second part investigates another geometric partition problem, related to planar representation of graphs. The slope number of a graph G is the smallest number s with the property that G has a straight-line drawing with edges of at most s distinct slopes and with no bends. We examine the slope number of bounded degree graphs. Our main results are that if the maximum degree is at least 5, then the slope number tends to infinity as the number of vertices grows but every graph with maximum degree at most 3 can be embedded with only five slopes. We also prove that such an embedding exists for the related notion called slope parameter. Finally, we study the planar slope number, defined only for planar graphs as the smallest number s with the property that the graph has a straight-line drawing in the plane without any crossings such that the edges are segments of only s distinct slopes. We show that the planar slope number of planar graphs with bounded degree is bounded.Comment: This is my PhD thesi

    Bitangents of tropical plane quartic curves

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    We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic.Comment: 13 pages, 9 figures. Minor revisions; accepted for publication in Mathematische Zeitschrif

    Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends

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    We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two pairs of crossing edges share two vertices. We study the relations of these beyond-planar graph classes (beyond-planar graphs is a collective term for the primary attempts to generalize the planar graphs) to right-angle crossing (RAC) graphs that admit compact drawings on the grid with few bends. We present four drawing algorithms that preserve the given embeddings. First, we show that every nn-vertex NIC-planar graph admits a NIC-planar RAC drawing with at most one bend per edge on a grid of size O(n)×O(n)O(n) \times O(n). Then, we show that every nn-vertex 1-planar graph admits a 1-planar RAC drawing with at most two bends per edge on a grid of size O(n3)×O(n3)O(n^3) \times O(n^3). Finally, we make two known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at most one bend per edge and for drawing IC-planar RAC graphs straight-line
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