Fan-crossing free graphs and their relationship to other beyond-planar graphs

A graph is fan-crossing free if it has a drawing in the plane so that each edge crossed by independent edges, crossing edges have distinct vertices. On other hand, common vertex, they form fan. Both are prominent examples for beyond-planar graphs. Further well-known classes k-planar, k-gap-planar, quasi-planar, and right angle We use subdivision, node-to-circle expansion path-addition operations to distinguish all these classes. In particular, we show 2-subdivision of any free, which does not hold k-(gap)-planar graphs, respectively. Thereby, obtain graphs neither nor k-(gap)-planar. Finally, some unique embedding, there thinned maximal recognition problem NP-complete.