1. Beyond planarity of graphs Eyal Ackerman University of Haifa and Oranim College

2. Drawing graphs in the plane  Consider drawings of graphs in the plane s.t.  No loops or parallel edges  Vertices  distinct points  Edges  Jordan arcs (no self-intersection)  Two edges intersect finitely many times  Intersection = crossing / common vertex  No three edges cross at a point  Topological graphs  Two edges intersect at most once  Simple topological graphs  Straight-line edges  Geometric graphs

4. The Crossing Lemma

6. Applications

7. Applications: Albertson Conjecture

9. The local (pair) crossing number

10. A Hanani-Tutte-type problem

16. Lower bounds

17. Virtually crossing edges parallel / avoiding edges virtually crossing edges

18. Virtually crossing edges parallel / avoiding edges virtually crossing edges

19. Virtually crossing edges (2)

20. Fan-planar graphs * that’s actually part of the definition of fan-planar graphs there and elsewhere

21. Fan-planar graphs (2)

22. Yet another not-far-from-planar graph

23. Thank you