Planarity and Higher Order Embeddings

Shawn CoxCS 594: Graph Theory

3-5-2014

Gas – Water – Electric Problem

Three Rivals all want service from the three utility services.

If any of the service lines cross, there will be a fight between the rivals

Is it possible for the utilities to lay the connections out so that no confrontations arise?

Gas – Water – Electric Problem

Turn the problem into a graph theory problem.– Rivals and Utilities are Vertices– Connections between them are

Edges– Can we draw it without crossing?

Planarity - Definition

In a given drawing of a graph on a plane, we can count the number of times that edges that cross at non-vertex points.

We can only consider crossings of exactly two edges

# of crossings: 1 # of crossings: 3# of crossings: Undefined

Planarity - Definition

For any given graph, there any number of representations, each with a possibly different number of crossings.

The smallest such number is the crossing number of a graph.

Planarity – Definition

A graph G is said to be a planar graph if it has crossing number 0.

Planarity – Examples

Planarity – Non-Examples

Not all graphs can be redrawn so that they are planar.

The smallest non-planar graphs are K5 and K3,3.

Euler’s Characteristic for Planar Graphs V – E + F = 2– V = # of vertices in the graph– E = # of edges in the graph– F = # of faces in the graph

Used to describe the shape of a graph

Euler’s Characteristic

Only holds for planar graphs Unable to determine faces if

intersecting edges are allowed

Disproving Planarity – K5

K5 has 5 vertices and 10 edges.

If K5 was planar, it would have 7 faces.

A planar graph with n vertices has at most 2/3 as many faces as it does edges–Why? Every edge is on exactly two faces,

and every face has at least 3 edges on its boundary.

Seven faces is one face too many.

Disproving Planarity – K3,3

K3,3 has 6 vertices and 9 edges.

If K3,3 was planar, it would have 5 faces.

A planar graph with n vertices that has no triangles has at most ½ as many faces as it does edges.– Similar to before, every edge is used twice,

and every face has at least 4 edges on its boundary.

Once again, five faces is one too many.

Generalizing Arguments

Finding a disproof for every graph?

Instead, consider subgraphs of the original.

If there is exists a subgraph of G that is non-planar, then G is non-planar.– Alternatively, G is planar if all of its

subgraphs are planar.

Formalized Arguments

Kuratowski’s Theorem – 1930– Occasionally attributed to Lev

Pontryagin as well, a Soviet Mathematician that claimed to have proven the theorem, but never published a proof.

Wagner’s Theorem – 1937

Kuratowski’s Theorem

G is planar if and only if there is no subgraph of G that is a subdivision of K5 or K3,3. Any such subgraph is called a Kuratowski

Subgraph

Kuratowski’s Theorem

Wagner’s Theorem – 1937

G is planar if and only if there is no subgraph of G that is a minor of K5 or K3,3.

Published seven years following Kurtowski’s theorem

Equivalent to Kuratowski’s proof, since it is easy to convert the appropriate minor into a subdivision and vice versa.

Testing for Planarity

Tons of graphs, and a lot of them are planar.

Number of Vertices

Number of Graphs

Number of Planar Graphs

1 1 1

2 2 2

3 4 4

4 11 11

5 34 33

6 156 142

Testing for Planarity

Path Addition Method– Originally published by Hopcroft and

Tarjan in 1974 for an O(n) algorithm Vertex Addition Method– Published by Lempel, Even, and

Cederbaum in 1967 for a O(n2) algorithm

– Improved by Even and Tarjan and then Booth and Leuker to O(n)

– Outperforms Path Addition.

Results of Planarity

Draw it without crossing lines Draw the graph with only

straight lines (Fary’s Theorem) Chromatic number of at least 4

Higher Order Analogies

Is the sphere different from the plane?–No. In fact the ability to draw a graph on

one immediately gives you a drawing on the other.

What is different then?– The Torus provides a different space.• Certain graphs can be drawn with crossing

number 0 on a toroid, but not on a plane.• Any such graph on the toroid has at most

chromatic number 7 (Heawood Conjecture).

Genus 1 Embeddings We can extend the notion of planarity to

higher orders.– If we put a graph G on a toroid, we say that it

is embedded on the toroid if it has crossing number 0.

We can color any genus 1 embedding with at most 7 colors.– The following is the canonical example of a 7

colorable toroidal embedding.

Open Problems

Any planar graph can be drawn with straight lines only, but can it be done with straight lines of integer lengths? (Harborth)– Can do done for cubic graphs, but

not known for the general case.

Homework Problem 1

Prove that the Heawood Graph (given below) is non-planar.

Homework 2

Prove or Disprove: The following graph is planar.

Homework 3

Prove or Disprove: The following graph is planar.

References Kuratowski, Kazimierz (1930), "Sur le problème des

courbes gauches en topologie", Fund. Math. (in French) 15: 271–283.

Wagner, K. (1937), "Über eine Eigenschaft der ebenen Komplexe", Math. Ann. 114: 570–590, doi:10.1007/BF01594196.

Hopcroft, John; Tarjan, Robert E. (1974), "Efficient planarity testing", Journal of the Association for Computing Machinery 21 (4): 549–568, doi:10.1145/321850.321852.

Lempel, A.; Even, S.; Cederbaum, I. (1967), "An algorithm for planarity testing of graphs", in Rosenstiehl, P., Theory of Graphs, New York: Gordon and Breach, pp. 215–232.

References Even, Shimon; Tarjan, Robert E. (1976), "Computing an st-

numbering", Theoretical Computer Science 2 (3): 339–344, doi:10.1016/0304-3975(76)90086-4.

Boyer & Myrvold (2004), p. 243: “Its implementation in LEDA is slower than LEDA implementations of many other O(n)-time planarity algorithms.”

Fáry, István (1948), "On straight-line representation of planar graphs", Acta Sci. Math. (Szeged) 11: 229–233, MR 0026311.

Weisstein, Eric W. "Heawood Conjecture." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HeawoodConjecture.html