Article

Outerplanar graph drawings with few slopes

Computational Geometry (Impact Factor: 0.57). 05/2012; DOI: 10.1016/j.comgeo.2014.01.003
Source: arXiv

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

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Article: Drawings of planar graphs with few slopes and segments
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ABSTRACT: We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.
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Article: Bounded-degree graphs can have arbitrarily large slope numbers.
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ABSTRACT: A straight-line or geometric drawing of a graph G is a layout of G in the plane such that the vertices are represented by distinct points, the edges are represented by (possibly crossing) line segments connecting the corresponding point pairs and not passing though any other point that represents a vertex. The slope number of G is the smallest number of distinct edge slopes used in a straight-line drawing of G. The authors prove that, for any given integer d≥5, there exists an n-vertex graph of maximum degree d whose slope number is at least n 1/2-1/(d-2)-o(1) . In particular, bounded-degree graphs can have arbitrarily large slope number, solving an open problem posed by V. Dujmović, M. Suderman and D. R. Wood [Lect. Notes Comput. Sci. 3383, 122–132 (2005; Zbl 1111.68571)], and improving a result by J. Barát, J. Matoušek and D. R. Wood [Electron. J. Comb. 13, No. 1, Research paper R3 (2006; Zbl 1080.05063)].
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Article: Drawing planar graphs of bounded degree with few slopes
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ABSTRACT: We settle a problem of Dujmovi\'c, Eppstein, Suderman, and Wood by showing that there exists a function $f$ with the property that every planar graph $G$ with maximum degree $d$ admits a drawing with noncrossing straight-line edges, using at most $f(d)$ different slopes. If we allow the edges to be represented by polygonal paths with {\em one} bend, then $2d$ slopes suffice. Allowing {\em two} bends per edge, every planar graph with maximum degree $d\ge 3$ can be drawn using segments of at most $\lceil d/2\rceil$ different slopes. There is only one exception: the graph formed by the edges of an octahedron is 4-regular, yet it requires 3 slopes. These bounds cannot be improved.
SIAM Journal on Discrete Mathematics 09/2010; · 0.58 Impact Factor