On the rectilinear crossing number of complete balanced multipartite graphs and balanced layered graphs

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications Pub Date : 2025-04-28 DOI:10.1016/j.comgeo.2025.102199

Ruy Fabila-Monroy , Rosna Paul , Jenifer Viafara-Chanchi , Alexandra Weinberger

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引用次数: 0

Abstract

A rectilinear drawing of a graph is a drawing of the graph in the plane in which the edges are drawn as straight-line segments and its vertices are points in general position. The rectilinear crossing number of a graph is the minimum number of pairs of edges that cross over all rectilinear drawings of the graph. Let

n \geq r

be positive integers. The graph

K_{n}^{r}

, is the complete balanced r-partite graph on n vertices, in which every set of the partition has at least

⌊ n / r ⌋

vertices. The balanced layered graph,

L_{n}^{r}

, is an r-partite graph on n vertices, where n is multiple of r. Every partition of

L_{n}^{r}

contains

n / r

vertices; for every

1 \leq i \leq r - 1

, all the vertices in the i-th partition are adjacent to all the vertices in the

(i + 1)

-th partition, and these are the only edges of

L_{n}^{r}

. In this paper, we give upper bounds on the rectilinear crossing numbers of

K_{n}^{r}

and

L_{n}^{r}

查看原文本刊更多论文

完全平衡多部图与平衡层图的直线交叉数

图形的直线绘制是在平面上绘制图形，其中的边绘制为直线段，其顶点是一般位置上的点。图的直线相交数是在图的所有直线图上相交的最小边对数。设n≥r为正整数。图Knr是n个顶点上的完全平衡r部图，其中每一个分区集至少有⌊n/r⌋顶点。平衡层图Lnr是一个有n个顶点的r部图，其中n是r的倍数，Lnr的每一个划分包含n/r个顶点；对于每一个1≤i≤r−1，第i个分区中的所有顶点与(i+1)-第i个分区中的所有顶点相邻，并且这些是Lnr的唯一边。本文给出了Knr和Lnr的直线相交数的上界。

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.