1 research outputs found
Fan-Crossing Free Graphs and Their Relationship to other Beyond-Planar Graphs
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 -planar, -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 -(gap)-planar graphs,
respectively. Thereby, we obtain graphs that are fan-crossing free and neither
fan-crossing nor -(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.Comment: 23 pages, 10 figure