Planarity Cox, Sherman, Tong

Key Concepts

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

Planarity

Planarity

Subdivisions

Edge Contraction

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

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′).

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.

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

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

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

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

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.

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

Robertson-Seymour Theorem Quasi-orders graphs based on their component minors, providing a tool for proofs. Image Source: http://www.theoremoftheday.org/CombinatorialTheory/Robertson-Seymour/TotDRobertsonSeymour.pdf

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.

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

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

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

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.

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.

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

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

Open Questions Harborth’s Conjecture Matchstick Graphs

References Planarity Testing Moris, Ondrej http://www.fi.muni.cz/~hlineny/Teaching/AGTT/2008/planarity-testing.pdf http://www.theoremoftheday.org/ ALGORITHM OF DEMOUCRON, MALGRANGE, PERTUISET, Kohnert, Axle http://www.mathe2.uni-bayreuth.de/EWS/demoucron.pdf Graph Algorithms http://en.wikipedia.org/wiki/Book:Graph_Algorithms

Questions?

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

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

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