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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
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