当前位置: X-MOL 学术arXiv.cs.CG › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Fan-Crossing Free Graphs
arXiv - CS - Computational Geometry Pub Date : 2020-03-18 , DOI: arxiv-2003.08468
Franz J. Brandenburg

A graph is fan-crossing free if it admits a drawing in the plane so that each edge can be crossed by independent edges. Then the crossing edges have distinct vertices. In complement, a graph is fan-crossing if each edge can be crossed by edges of a fan. Then the crossing edges are incident to a common vertex. Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing is assigned to an edge involved in the crossing, so that at most k crossings are assigned to each edge. We use the s-subdivision, path-addition, and node-to-circle expansion operations to show that there are fan-crossing free graphs that are not fan-crossing, k-planar, and k-gap-planar for k >= 1, respectively. A path-addition adds a long path between any two vertices to a graph. An s-subdivision replaces an edge by a path of length s, and a node-to-circle expansion substitutes a vertex by a 3-regular circle, so that each vertex of the circle inherits an edge incident to the original vertex. We introduce universality for an operation and a graph class, so the every graph has an image in the graph class. In particular, we show the fan22 crossing free graphs are universal for 2-subdivision and for node-to-circle 3 expansion. Finally, we show that some graphs have a unique fan-crossing free embedding, that there are maximal fan-crossing free graphs with less edges than the density, and that the recognition problem for fan-crossing free graphs is NP-complete.

中文翻译:

风扇交叉自由图

如果一个图允许在平面中进行绘图,使得每条边都可以被独立的边穿过,那么它就是无扇形交叉的。然后交叉边有不同的顶点。作为补充,如果每条边都可以被扇形边穿过,则图是扇形交叉的。然后交叉边与公共顶点相交。如果每条边最多与 k 条边交叉,则图是 k 平面的,如果每个交叉点都分配给参与交叉的边,则图为 k 平面,因此每条边最多分配 k 个交叉点。我们使用 s-subdivision、path-addition 和 node-to-circle 展开操作来证明对于 k >= 存在不是扇形交叉、k-planar 和 k-gap-planar 的扇形交叉自由图1、分别。路径添加将任意两个顶点之间的长路径添加到图形中。s 细分用长度为 s 的路径替换边,并且节点到圆的扩展将顶点替换为 3 正则圆,因此圆的每个顶点都继承了与原始顶点相关的边。我们为一个操作和一个图类引入了通用性,因此每个图在图类中都有一个图像。特别是,我们展示了 fan22 交叉自由图对于 2 细分和节点到圆 3 扩展是通用的。最后,我们证明了一些图具有独特的扇形交叉自由嵌入,存在边比密度少的最大扇形交叉自由图,并且扇形交叉自由图的识别问题是 NP 完全的。所以每个图在图类中都有一个图像。特别是,我们展示了 fan22 交叉自由图对于 2 细分和节点到圆 3 扩展是通用的。最后,我们证明了一些图具有独特的扇形交叉自由嵌入,存在边比密度少的最大扇形交叉自由图,并且扇形交叉自由图的识别问题是 NP 完全的。所以每个图在图类中都有一个图像。特别是,我们展示了 fan22 交叉自由图对于 2 细分和节点到圆 3 扩展是通用的。最后,我们证明了一些图具有独特的扇形交叉自由嵌入,存在边比密度少的最大扇形交叉自由图,并且扇形交叉自由图的识别问题是 NP 完全的。
更新日期:2020-03-20
down
wechat
bug