Beyond planarity of graphs - PowerPoint Presentation

Slide1

Beyond planarity of graphs

Eyal

Ackerman

University of Haifa and

Oranim

CollegeSlide2

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

graphsSlide3

-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

 Slide4

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 .  Slide5

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  Slide6

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  Slide7

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, .  Slide8

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]. Slide9

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

.

 Slide10

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 ? Slide11

Decomposing

-planar graphs

Every

-plane graph can be decomposed into

plane graphs:

Remove a maximal plane subgraph

Yields -plane graphRecall:-plane graph = plane + forest [A. ‘14] Problem: -plane graph = plane + forest ?What about -plane graphs for ? max size of a -plane graphSlide12

-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 noA 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. Slide13

-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 constantsMax 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. Slide14

-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. Slide15

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] Slide16

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 edgesAny planar graph can be embedded into any set of points according to any bijection [Pach & Wenger ‘01] for geometric graphs:   Slide17

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 edgesvirtually crossing edgesSlide18

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-planarityProblem: 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 edgesvirtually crossing edgesSlide19

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] Slide20

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 trueHolds 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 elsewhereSlide21

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.

 Slide22

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  Slide23

Thank you