1. Planarity
Cox, Sherman, Tong

2. Key Concepts

3. Terminology Crossing Number
The minimum number of edges that cross in any drawing of a graph.
Planarity
We call a graph planar if it has crossing number 0 (i.e. can be drawn without edges crossing).
Subdivision
Replace an edge with a degree two vertex connected to the original adjacent vertices
Edge-Contraction
Merging two adjacent vertices into a single vertex of higher degree

4. Planarity

5. Planarity

6. Subdivisions

7. Edge Contraction

8. Minors Several types of subgraphs to consider:
Ordinary Subgraphs: Standard concept of subgraph
Topological Minors: A subgraph with subdivided edges
Graph Minors: A subgraph with contracted edges

9. Minors
A graph X is a minor of G if it can be constructed by contracting edges from G.
More formally, X is a minor of Y if and only if there is a map ϕ from a subset of V (Y ) onto V (X) such that for every vertex x ∈ X its inverse image ϕ−1 (x) is connected in Y and for every edge xx′∈X there is an edge in Y between the sets ϕ−1(x) and ϕ−1(x′).

10. Minors Minors are a generalization of subgraph isomorphism.
If X is a subgraph isomorphism of G -> X is a minor of G.
The reverse does not hold.

11. Forbidden Minors
Forbidden minors are used to classify many categories of graphs, including planar graphs. K3,3 and K5 are forbidden minors for planarity.

12. Forbidden Characterizations
Graph Family
Forbidden Minor
Forest K3
Planar
K3,3, K5

13. Euler’s Characteristic
A planar graph must satisfy: v-e+f=2
Example:
e=6, v = 5, f = 3 or
e=16, v = 15, f = 3
Also, if e > 3v-6, then the graph
cannot be planar

14. Fary’s Theorem
Fary’s Theorem states that using curved lines does not increase the number of planar graphs which can be drawn.

15. Kuratowski’s Theorem
Kuratowski’s theorem is that any planar graph cannot have the graphs K3,3 or K5 embedded in it as a topological minor.

16. Wagner’s Theorem
Wagner’s Theorem characterizes planar graphs by their minors. It goes beyond Kuratowski’s theorem by using graph minors rather than topological minors

17. Robertson-Seymour Theorem
Quasi-orders graphs based on their component minors, providing a tool for proofs.
Image Source:

18. Robertson-Seymour Theorem
The set of finite graphs is well-quasi-ordered by graph minors.
Given graphs G and H and H ≤ G meaning H is a graph minor of G: H is a subgraph of G with some edges contracted. Let P be a property of graphs (such as the property of being planar) which is closed under graph minors: G has property P and H ≤ G means that H also has property P. Now let F be a family of graphs failing to have property P. Suppose F is minor minimal: any minor of any graph in F does have P, then F must be finite. For, if not, list the graphs of F : x0, x1, By Robertson-Seymour, F contains graphs x and y with x ≤ y. Then x has property P by the assumption that F was minor-minimal, but x ∈ F means that x does not have property P. This contradiction means that F cannot be infinite. QED.

19. Demoucron, Malgrange and Pertuiset Algorithm
(1) choose a cycle of G this is a planar graph G’ together with a embedding
(2) compute all faces of G’
(3) compute F(S) = set of fragments of G with respect to G’
(4) if F(S)=Ø then we have G’∼=G and G’ has a planar embedding. end
(5) compute all admissible faces for all fragments
(6) if there is a fragment without admissible face, then the graph has no planar embedding. end
(7) if there is a fragment S with only one admissible face then goto 9
(8) choose a fragment S (more than one admissible face)
(9) choose a α-path from S and embed it into an admissible face of S, goto 2

20. Lempel-Even-Cederbaum Algorithm
1: find st-order v1, , vn of the vertices of G
2: G1 ← ({v1}, ∅)
3: i ← 1
4: while Gi != G do
5: find bush form Bi for Gi
6: if ∃ a seq. of perm. and rev. making all extra vert. i + 1 consecutive then
7: proceed such transformation and add vertex vi+1 to obtain Gi+1
8: i ← i + 1
9: else
10: return False {by Theorem 3}
11: end if
12: end while
13: return True

21. Number of Planar Graphs
Number of Vertices
Number of Distinct Graphs
Number of Distinct Planar Graphs 1 2 3 4 11 5 34 33 6
156
142 7
1044
822

22. Higher Order Spaces What about spherical graphs?
Embedding a graph on a sphere is equivalent to embedding on a plane.
Both the sphere and the plane are Genus 0 surfaces.

23. Higher Order Spaces What are higher order spaces?
The Torus has genus one and can be used to embed much more complex graphs.
By adding more holes to our surface, we increase the genus and the potential for embedded graphs.

24. Example Torus Embedding
The Heawood graph cannot be embedded in the plane (it has crossing number 3).

25. Example Torus Embedding
However, the Heawood Graph can be embedded around a torus.

26. Open Questions
Harborth’s Conjecture
Matchstick Graphs

27. References
Planarity Testing Moris, Ondrej
ALGORITHM OF DEMOUCRON, MALGRANGE, PERTUISET, Kohnert, Axle
Graph Algorithms

28. Questions?

29. Homework Question 1
The Petersen graph contains both forbidden minors within it. Demonstrate how to obtain either minor through subdivisions and/or edge contractions.

30. Homework Question 2
The provided drawing of the Heawood graph has crossing number 14. Redraw the graph with crossing number 3 instead.

31. Homework Question 3
Decide whether the given graph is planar or not.