# On Fan-Crossing Graphs

A fan is a set of edges with a single common endpoint. A graph is fan-crossing if it admits a drawing in the plane so that each edge is 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 graph is adjacency-crossing if it admits a drawing so that crossing edges are adjacent. Then it excludes independent crossings which are crossings by edges with no common endpoint. Adjacency-crossing allows triangle-crossings in which an edge crosses the edges of a triangle, which is excluded at fan-crossing graphs. We show that every adjacency-crossing graph is fan-crossing. Thus triangle-crossings can be avoided. On the other hand, there are fan-crossing graphs that are not fan-planar, whereas 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 their density with at most 5n - 10 edges for graphs of size n.

research
03/18/2020

### Fan-Crossing Free Graphs

A graph is fan-crossing free if it admits a drawing in the plane so that...
research
09/12/2023

### The maximum size of adjacency-crossing graphs

An adjacency-crossing graph is a graph that can be drawn such that every...
research
03/13/2023

### Drawings of Complete Multipartite Graphs Up to Triangle Flips

For a drawing of a labeled graph, the rotation of a vertex or crossing i...
research
09/21/2018

### Polyline Drawings with Topological Constraints

Let G be a simple topological graph and let Γ be a polyline drawing of G...
research
08/24/2020

### Limiting crossing numbers for geodesic drawings on the sphere

We introduce a model for random geodesic drawings of the complete bipart...
research
08/25/2021

### Star-Struck by Fixed Embeddings: Modern Crossing Number Heuristics

We present a thorough experimental evaluation of several crossing minimi...
research
12/15/2020

### Edge-Minimum Saturated k-Planar Drawings

For a class 𝒟 of drawings of loopless multigraphs in the plane, a drawin...