1. Algorithms and Networks
Planar graphs
Algorithms and Networks

2. Planar graphs Can be drawn on the plane without crossings
Plane graph: planar graph, given together with an embedding in the plane
Many applications…
Questions:
Testing if a graph is planar
Finding a nice or good drawing of a planar graph
But what does that mean?
Solving problems for planar graphs
Planar graphs

3. Graph drawing / network layout
Applications:
Visually representing a network (e.g., social network, organization structure, data bases (ER-diagrams), software (e.g., UML-diagrams), flow charts, phylogenetic trees (biology, evolution), …
Design of “chip” layout (VLSI) …
Planar graphs

4. Some notions Faces Exterior face Interior faces
Each edge is incident to 2 faces, except in special cases:
Planar graphs

5. The smallest graphs that are not planar
K5, K3,3
Planar graphs

6. Kuratowski / Wagner
Theorem (Kuratowski, Wagner, 193*) A graph is planar, if and only if it does not contain the K5 and the K3,3 as a homeomorphic subgraph / as a minor.
H is a minor of G, if H can be obtained from G by a series of 0 or more deletions of vertices, deletions of edges, and contraction of edges.
Does not yield fast recognition algorithm!
Planar graphs

7. The Petersen graph is not planar, as it has K3,3 as minor
Animation from Wikipedia
Planar graphs

8. Euler’s theorem
Theorem (Euler) Let G be a connected plane graph with n vertices, m edges, and f faces. Then n + f – m = 2.
Proof. By induction.
True if m=0.
If G has a circuit, then delete an edge and …
If G has a vertex v of degree 1, then delete v and … …
Planar graphs

9. Euler’s theorem Corollaries
If G is a connected plane graph with no parallel edges and no self-loops, with n > 1, then m £ 3n-6.
Every face `has’ at least three edges; each edge `is on’ two faces, or twice on the same face.
Every plane graph with no parallel edges and no self-loops has a vertex of degree at most 5.
Has algorithmic consequences…
Planar graphs

10. Duality The dual G* of a plane graph G
A vertex in G* for each face of G
An edge in G* when faces share an edge in G
Planar graphs

11. Drawings of planar graphs
Different types
Vertices are:
Points in 2-dimensional space
Rectangles, other objects
Edges are
Straight lines
Curves
Lines with bends
Adjacencies or intersections of objects
Usual model
Planar graphs

12. Drawing planar graphs In steps: Test if G is planar, and
Find for each vertex, a clockwise ordering of its incident edges, such that these orderings allow a planar embedding, and then
Assign coordinates to vertices
Planar graphs

13. Planarity testing: reduction to biconnected graphs
First, suppose G is biconnected.
G is biconnected, if for each vertex v, G-v is connected.
A biconnected component of G is a maximal subgraph that is biconnected.
A graph is planar, if and only if each of its biconnected components is planar.
Each biconnected component can be drawn with any vertex on its exterior face.
Build drawing of G by putting drawings of biconnected components together.
Easier non-constructive argument: G has a K5 or K3,3 minor, iff a biconnected component of G has such a minor.
Planar graphs

14. Cycles with pieces Take a cycle C in G.
Divide edges not on C of G into classes:
e and e’ in the same class if there is a path starting with e, ending with e’, using no vertices on C as intermediate points
A class forms a piece.
Pieces are
A single edge between two vertices on C
A connected graph with at least one vertex not on C
Planar graphs

15. Separating cycles A cycle is separating if it has at least two pieces.
Plan:
Find separating cycle C (how?)
Test for each piece + C if it is planar (recursion)
Determine for each piece if it has to be drawn inside or outside C (matching problem; details follow).
Planar graphs

16. Finding a separating cycle
Find a cycle C.
If C separating: done.
Otherwise, let P be the unique piece.
If P is a path: G is planar: done.
Otherwise:
Take two consecutive attachments v, w of P in the circular ordering of C, and let Q be the part of C between the attachments without another attachment.
Take a path R between v and w in P avoiding other attachments
C – Q + R is a separating cycle
Planar graphs

17. The interlacement graph
Pieces can be `inside’ or `outside’ the cycle
Build interlacement graph
Each piece is a vertex in interlacement graph
Pieces are adjacent, when they cannot be at the same side (look to attachments)
Planar graphs

18. Theorem
Let C be a separating cycle in G. G is a planar graph, if and only if
For each piece P, C+P is planar.
The interlacement graph is bipartite.
Gives a polynomial
time algorithm!
Planar graphs

19. Interlacement graph can be built in O(n2) time
For each piece
Partition C into intervals with respect to attachments
For each other piece
Look whether all attachments are in the same interval
If not, they conflict.
Planar graphs

20. Algorithm Find a cycle C Test if C is separating.
If not, use method to build separating cycle C’.
Find pieces of C’.
Build interlacement graph.
Test if interlacement graph is bipartite. If not: G is not planar, stop.
Otherwise, test for each piece P, recursively, if P+C’ is planar.
Planar graphs

21. Time O(n3). Can be done faster: O(n) algorithms exist.
Finding clockwise orderings of edges per vertex can also be done: several details…
Planar graphs

22. Force directed methods
Method for arbitrary graphs
Uses physical analogy:
Force system
Method to find equilibrium state
Advantages:
Relatively easy to understand and implement
Often reasonably nice results
Planar graphs

23. Springs and electrical forces method
Edges are springs
Vertices repel each other
Force on vertex:
fuv is force on spring
guv is repelling force
Planar graphs

24. Spring force Assume some `wanted length’ of an edge luv. (E.g., 1.)
Spring wants to assume length luv; force is proportional to difference with luv.
Choose some stiffness of the spring kuv.’
Force in x-direction is (y-direction similar):
Planar graphs

25. Similar in y-direction
Repelling force
Vertices are repelling particles
Follow inverse square law
k’uv is strength of repulsion (choose some number, e.g., experimentally)
Repelling force of v in x-direction:
Similar in y-direction
Planar graphs

26. Search for equilibrium
Position where all vertices have a force of 0 towards them (sum of spring and repelling forces)
Springs have lengths of edges go to desired length
Repelling has vertices not too close together
Planar graphs

27. Searching for equilibrium
Numerical techniques
Simple algorithm:
Take for each vertex a random position
Repeat:
Compute for each vertex its force
Move all vertices to new position depending on their forces
Until in sufficient rest
Faster algorithms exist
Planar graphs

28. Barycenter method Tutte, 1960 Differences with force method
No repelling forces
luv = 0
Stiffness is 1
Vertices at exterior face get fixed positions
Planar graphs

29. Where pu is the position of u on the plane
Force on a vertex
Where pu is the position of u on the plane
If v is a free vertex
Planar graphs

30. Finding equilibrium
System of forces can be written as set of linear equations
Fast converging algorithms
For planar graphs, system can be solved in O(n1.5) time
Planar graphs

31. Barycenter draw pseudocode
Place all fixed vertices at corners of polygon around origin, and all free vertices on origin
Repeat
For each free vertex v do
x(v) = 1/degree(v) * S((u,v) in E) x(u)
y(v) = 1/degree(v) * S((u,v) in E) y(u)
until sufficient convergence
Planar graphs

32. Theorem of Tutte
If G is a triconnected planar graph, then this method yields a planar drawing of G.
Assuming …
In practice, layers become smaller in the inside
Generalizations with more complicated forces sometimes also have this property
If G is not triconnected, then we can make it triconnected by adding edges
Planar graphs

33. Grid drawings Many results on planar drawings, e.g.,
Theorem (Schnyder, 1990) Every planar graph G has a plane drawing, such that
Each edge is a straight line
Vertices are mapped to pairs of integers in {0,1,…,n – 2} * {0,1, …, n – 2}
Planar graphs

34. Many other results Embeddings with small areas and few bends per edge
Drawings where vertices are represented by rectangles with an edge if rectangles touch …
Planar graphs