Fan-Crossing Free Graphs and Their Relationship to other Beyond-Planar Graphs

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