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

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

The Crossing Lemma

Applications

Applications: Albertson Conjecture

The local (pair) crossing number

A Hanani-Tutte-type problem

Lower bounds

Virtually crossing edges parallel / avoiding edges virtually crossing edges

Virtually crossing edges parallel / avoiding edges virtually crossing edges

Virtually crossing edges (2)

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

Fan-planar graphs (2)

Yet another not-far-from-planar graph

Thank you