1. Great Theoretical Ideas In Computer Science
Planar Graphs
Lecture 23 CS

2. A graph is called planar if it can be drawn in the plane in such a way that no two edges cross.
Example of a planar graph:
The clique on 4 nodes.

3. Is K5 planar?

5. What about K3,3 ?

6. What about the intriguing object that I brought to class today?

7. The problem of drawing a graph in the plane arises frequently in VLSI layout problems.

8. Definition: When a graph is drawn in the plane with no crossed edges it is called a plane graph.
A plane graph cuts the plane into regions that we call faces.
one face
two faces

9. Question:
Can you redraw this graph as a plane graph so as to alter the number of its faces?

10. This graph
has
6 vertices
8 edges
and 4 faces
vertices – edges + faces = 2

11. This graph
has
7 vertices
12 edges
and 7 faces
vertices – edges + faces = 2

12. Euler 1752 If G is a connected plane graph, then
vertices – edges + faces = 2
Let v = # of vertices
e = # of edges
f = # of faces

13. Proof: By induction on the # of cycles of G.
Base case: G has no cycles.
G is connected so it must be a tree. Thus, e = v - 1 and f = 1.

14. Suppose G has at least one cycle C containing edge e.
Let
v= # of vertices, e = # of edges,
f  = # of faces
exterior e
interior
G is connected since e was on a cycle.
f = f-1 and G has fewer cycles than G.
v= v e= e-1
By induction hypothesis:

15. Corollary:
No matter how we redraw a plane graph it will have the same # of faces.
Proof:
f = 2 – v + e is determined by v and e, neither of which change when we redraw the graph.

16. Platonic Solids
A Platonic solid has congruent regular polygons as faces and has the same number of edges meeting at each corner.
Each one can be flattened into a planar graph:
With constant degree: k
and the same number of edges bounding each face: l

17. = Each edge belongs to 2 faces: By Euler’s formula:
# of edges
coming from n
vertex n =
Each edge belongs to 2 faces:
By Euler’s formula:
and k,l  3 for physical reasons

18. The only solutions:
tetrahedron
cube
octahedron
dodecahedron
icosahedron

19. Theorem: Every (simple) n-node planar graph G has at most 3n-6 edges.
Proof: n = 3: Clearly true.
n  3: consider a graph G with a maximal number of edges.
G must be connected or else we could add an edge.
Thus
Every face has at least 3 edges on its boundary.
Thus
Every edge lies on the boundary of at most 2 faces.

20. The Kuratowski Graphs

21. Corollary: K5 is not planar.
A planar graph on n = 5 nodes can have at most 3n-6 = 9 edges.
Thus: K5 is not planar.

22. Fact: K3,3 is not planar either.x y b c z
When we redraw K3,3 , the yellow cycle will be laid out: a b c x y z

23. Insight 1.
If we replace edges in a Kuratowski graph by paths of whatever length, they remain non-planar.

24. Insight 2
If a graph G contains a subgraph obtained by starting with K5 or K3,3 and replacing edges with paths, then G is non-planar.

25. Kuratowski’s Theorem [1930]
A graph is planar if and only if it contains no subgraph obtainable from K5 or K3,3 by replacing edges with paths.

26. Appel-Haken Four-Color Theorem [1976]
The vertices of any planar graph can be 4-colored in such a way that no two adjacent vertices receive the same color.

27. User Interface How many objects appear in the pink window?
(real-time response required) 2 4 3

28. Idea: View each rectangle as a mesh.
(mesh is planar)
interior
faces

29. Objects are placed on an n  n grid.
window border touches no nodes
Data structure will contain:
for each grid node
# objects containing node
for each grid edge
# objects containing edge
for each (interior) face
#objects containing face

30. Partial Overlap of Window and Object
ignore object’s nodes and edges outside window

31. # of objects intersecting
node counts in
edge counts in
face counts in
# of objects intersecting

32. Inclusion-Exclusion Speed-up!
store
3-tuple:
sum of node counts in
sum of edge counts in
sum of face counts in

33. a b c d
Node count in is a – b – c + d. d a b c 1 -1