Planarity and Higher Order Embeddings Shawn Cox CS 594: Graph Theory

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 K 5 and K 3,3.

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 – K 5  K 5 has 5 vertices and 10 edges.  If K 5 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 – K 3,3  K 3,3 has 6 vertices and 9 edges.  If K 3,3 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 K 5 or K 3,3.  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 K 5 or K 3,3.  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

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(n 2 ) 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: /BF  Hopcroft, John; Tarjan, Robert E. (1974), "Efficient planarity testing", Journal of the Association for Computing Machinery 21 (4): 549–568, doi: /  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: / (76)  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  Weisstein, Eric W. "Heawood Conjecture." From MathWorld-- A Wolfram Web Resource.