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.
A graph is 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 isfan-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.
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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,
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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
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with less edges than the density