Beyond planarity of graphs

Eyal AckermanUniversity 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

-planar graphs A topological graph is -plane if every edge is crossed at

most times. A graph is -planar if it can be drawn as a -plane

topological graph. = max size of a -planar graph

[Pach and Tóth ‘97] [Pach et al. ‘06] [A. ‘14]

Problem: determine . by the Crossing Lemma

The Crossing Lemma

Crossing Lemma: For every graph with vertices and edges .

[Ajtai, Chvátal, Newborn, Szemerédi ’82; Leighton ‘83]

= crossing number = minimum number of crossings in a drawing of .

The Crossing Lemma

Crossing Lemma: For every graph with vertices and edges .

[Ajtai, Chvátal, Newborn, Szemerédi ’82; Leighton ‘83]

Tight apart from . Originally , later [Pach & Tóth ‘97]: [Pach et al. ‘06]: [A. ‘14]:

Problem [Tóth, Emléktábla 2011]: improve the bounds on . [Pach & Tóth ‘97]

Using new bounds on for small

Applications Improving the crossing lemma yields immediate

improvements in all of its applications. For example: , for

• Previous best constant factor was

The number of incidences between lines and points in the plane is at most .

• Previous best constant factor was • Should be greater than

Applications: Albertson Conjecture Albertson Conj.: if then . It suffices to verify for -critical graphs – trivial Four Color Theorem:

Suppose there is a planar graph with . However, by AC .

If then cannot be planar, .

Applications: Albertson Conjecture Albertson Conj.: if then . It suffices to verify for -critical graphs – trivial Four Color Theorem [Oporowskia & Zhao ‘09]

[Albertson, Cranston & Fox ‘10]

[Barát & Tóth ‘10]

[A. ‘14] following [Barát & Tóth ‘10]. AC holds for -vertex -critical graphs if or [A. ‘14, Barát & Tóth

‘10].

The local (pair) crossing number = min s.t. is -planar = min s.t. is can be drawn such that every edge crosses

at most other edges (possibly more than once). Clearly, . s.t. :

[Schaefer & Štefankovič 2004]

If then [A. & Schaefer ‘14]. Problem: Does imply ?

If true, then implies . Can only show that implies .

A Hanani-Tutte-type problem Hanani-Tutte Thm: if can be drawn such that every

edge crosses no other edge an odd number of times, then is planar.

Problem: Is it true that if can be drawn such that every edge crosses at most one other edge an odd number of times, then is -planar? Can we show that is -planar for some ?

Decomposing -planar graphs Every -plane graph can be decomposed into plane

graphs: Remove a maximal plane subgraph Yields -plane graph

Recall:

-plane graph = plane + forest [A. ‘14] Problem: -plane graph = plane + forest ?

What about -plane graphs for ?

max size of a -plane graph

-quasi-planar graphs A topological graph is -quasi-plane if it has no pairwise

crossing edges. E.g.,-quasi-plane = plane graph

and -quasi-plane means no

A graph is -quasi-planar if it can be drawn as a -quasi-plane topological graph.

Conj.: For any every -quasi-planar graph has at most edges.

-quasi-planar graphs (2) Conj.: For any every -quasi-planar graph has at most

edges. Trivial for For :

[Agarwal et al. ‘97]: true for simple topological graphs [Pach et al. ‘03]: true for the general case [A. & Tardos ‘07]: simpler proofs with better constants

• Max size of a simple -quasi-planar graph is • Max size of a -quasi-planar graph is between and

For the conjecture holds [A. ‘09]. For the conjecture is open.

-quasi-planar graphs (3) Conj.: For any every -quasi-planar graph has at most

edges. For the conjecture is open. Best upper bounds on the size of -quasi-planar graphs:

for simple graphs [Suk & Walczack ‘13]

[Fox & Pach ‘12]

Problem: improve these bounds.

Decomposing -quasi-plane graphs Problem: what is the minimum number s.t. any -vertex -

quasi-plane graph can be decomposed into plane graphs? If is -quasi-plane then [Palwik et al. ‘14] [Fox & Pach, ’12] for -monotone graphs [Suk, ’14]

Lower bounds Problem: find non-trivial lower bounds on the maximum

size of a -quasi-planar graph. by overlaying edge-disjoint triangulations

The thickness of is • Most planar subgraphs have edges

Any planar graph can be embedded into any set of points according to any bijection [Pach & Wenger ‘01]

for geometric graphs:

𝑛−(𝑘−1)

𝑘−1

Virtually crossing edges Consider two (independent) edges in a geometric graph:

Conj.: For any every geometric graph with no pairwise virtually crossing edges has at most edges.

[Valtr ‘98]: For any every geometric graph with no pairwise parallel edges has at most edges.

parallel / avoiding edges virtually crossing edges

Virtually crossing edges Consider two (independent) edges in a geometric graph:

Conj.: For any every geometric graph with no pairwise virtually crossing edges has at most edges. For holds by -quasi-planarity

Problem: provide different proofs (and better bounds) For the maximum size is

• Not so easy if is not in general position [A., Nitzan, Pinchasi ‘14]

parallel / avoiding edges virtually crossing edges

Virtually crossing edges (2) Showing that a complete geometric graph has pairwise

virtually crossing edges is easy:

… whereas, showing that a complete geometric graph has pairwise crossing edges is an open problem. Best bound is only [Aronov et al. ‘94]

Fan-planar graphs A (simple) topological graph is fan-planar if for every

three edges if and cross then they share a vertex.

Easy: if is fan-planar then Conj.: if is fan-planar then

Tight if true Holds if and must share a vertex on the same side

of [Kaufmann & Ueckerdt]*:

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

Fan-planar graphs (2) Can we rule out “triangle” crossing?

Note that has no further crossings If yes, then all edges crossing share the same vertex.

Yet another not-far-from-planar graph “Maximal” fan- and -plane graphs satisfy:

for every two crossing edges there is a crossing-free edge that connects endpoints of these edges.

Problem: what is the maximum size of a topological graph satisfying the above? -quasi-planar, hence at most Should be At least

Thank you