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On fan-crossing graphs
Theoretical Computer Science ( IF 0.9 ) Pub Date : 2020-07-15 , DOI: 10.1016/j.tcs.2020.07.002
Franz J. Brandenburg

A fan is a set of edges with a single common endpoint. A drawing of a graph in the plane is fan-crossing if each edge can only be crossed by edges of a fan. It is fan-planar if, in addition, the common endpoint is on the same side of the crossed edge. A drawing is adjacency-crossing if any two edges are adjacent if they cross the same edge. Then multiple independent crossings are excluded, in which an edge is crossed by at least two edges with no common endpoint. In adjacency-crossing drawings it is allowed that an edge crosses the edges of a triangle, which is excluded for fan-crossing drawings. A graph is fan-crossing (fan-planar, adjacency-crossing) if it admits a respective drawing.

We show that every adjacency-crossing graph is fan-crossing and that there are fan-crossing graphs that are not fan-planar. Moreover, for every fan-crossing graph there is a fan-planar graph on the same set of vertices and with the same number of edges. Hence, fan-crossing and fan-planar graphs are different, but they do not differ in the density with at most 5n10 edges for graphs of order n.



中文翻译:

在扇形交叉图上

风扇是具有单个共同的端点的边的集合。如果每个边缘只能与风扇的边缘相交,则平面中的图形绘图会呈扇形相交。此外,如果公共端点在交叉边缘的同一侧,则为扇形平面。如果任何两个边线都跨越同一边线,则它们就是邻接线。然后排除多个独立的交叉点,其中一条边线被至少两个没有共同端点的边线交叉。在相交的工程图中,允许边与三角形的边相交,这在扇形工程中不适用。如果图形接受相应的图形,则为扇形交叉(扇形平面,邻接交叉)。

我们表明,每个邻接交叉图都是扇形交叉的,并且有扇形交叉图不是扇形的。此外,对于每个扇形交叉图,在相同的一组顶点上以及具有相同数量的边的位置上都有一个扇形平面图。因此,扇形交叉图和扇形平面图是不同的,但是它们的密度最多不超过5ñ-10n阶图的边。

更新日期:2020-09-16
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