Straight-line drawings of 1-planar graphs

IF 0.4 4区计算机科学 Q4 MATHEMATICS Computational Geometry-Theory and Applications Pub Date : 2023-07-04 DOI:10.1016/j.comgeo.2023.102036

Franz J. Brandenburg

引用次数: 0

Abstract

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. However, there are 1-planar graphs that do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line drawing with a two-coloring of the edges such that edges of the same color do not cross. Thus 1-planar graphs have geometric thickness two. In addition, the drawing is nearly 1-planar, that is, it is 1-planar if all fan-crossed edges are removed. An edge is fan-crossed if it is crossed by edges with a common vertex if it is crossed more than twice. The drawing algorithm uses high precision arithmetic with numbers with $O (n \log n)$ digits and computes the straight-line drawing from a 1-planar drawing in linear time on a real RAM.

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1-平面图的直线图

如果图可以在平面中绘制，使得每条边最多相交一次，那么它就是1-平面的。然而，有些单平面图不允许使用直线单平面图。我们证明了每个1-平面图都有一个边有两种颜色的直线图，这样同一颜色的边就不会相交。因此，1-平面图具有几何厚度2。此外，该图形几乎是单平面的，也就是说，如果删除了所有扇形交叉边，则该图形是单平面。如果一条边与具有公共顶点的边相交，如果该边相交两次以上，则该边为扇形相交。绘图算法使用高精度算术，数字为O（nlog⁡n）数字，并在真实RAM上以线性时间从1-平面图形计算直线图形。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Computational Geometry-Theory and Applications 数学-数学

CiteScore

1.60

自引率

16.70%

发文量

审稿时长

>12 weeks

期刊介绍： Computational Geometry is a forum for research in theoretical and applied aspects of computational geometry. The journal publishes fundamental research in all areas of the subject, as well as disseminating information on the applications, techniques, and use of computational geometry. Computational Geometry publishes articles on the design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects. Also welcomed are computational geometry solutions to fundamental problems arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographical information systems. Computational Geometry features a special section containing open problems and concise reports on implementations of computational geometry tools.

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