Outerplanar Graph Drawings with Few Slopes

Computational Geometry (Impact Factor: 0.48). 05/2012; 47(5). DOI: 10.1016/j.comgeo.2014.01.003
Source: arXiv


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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Available from: Kolja Knauer, Jul 29, 2014
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    • "This result goes in the direction of studying how many slopes may be needed to construct straight-line drawings that are " nearly-planar " . Moreover, since outerplanar drawings are a special case of the outer 1-planar drawings, this result generalizes the above mentioned upper bound on the (outer)planar slope number of outerplanar graphs [18]. Our results are constructive and give rise to linear-time drawing algorithms in the real RAM model of computation. "

    Preview · Article · Jan 2015 · Journal of Graph Algorithms and Applications
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    ABSTRACT: A graph is 1-planar if it can be embedded in the plane with at most one crossing per edge. It is known that the problem of testing 1-planarity of a graph is NP-complete. In this paper, we study outer-1-planar graphs. A graph is outer-1-planar if it has an embedding in which every vertex is on the outer face and each edge has at most one crossing. We present a linear time algorithm to test whether a given graph is outer-1-planar. The algorithm can be used to produce an outer-1-planar embedding in linear time if it exists.
    No preview · Article · May 2014 · Algorithmica
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    ABSTRACT: We consider drawings of graphs in the plane in which edges are represented by polygonal paths with at most one bend and the number of different slopes used by all segments of these paths is small. We prove that $\lceil\frac{\Delta}{2}\rceil$ edge slopes suffice for outerplanar drawings of outerplanar graphs with maximum degree $\Delta\geq 3$. This matches the obvious lower bound. We also show that $\lceil\frac{\Delta}{2}\rceil+1$ edge slopes suffice for drawings of general graphs, improving on the previous bound of $\Delta+1$. Furthermore, we improve previous upper bounds on the number of slopes needed for planar drawings of planar and bipartite planar graphs.
    Full-text · Article · Jun 2015