Planarity Testing

1
Planarity Testing
2
Overview

• Definitions
• Mengers Theorem
• Ear Decomposition Lemma
• Pieces
• Planarity Testing Algorithm
• Complexity O(n )

3
Well only discuss connected undirected graphs
3
Definitions

• Cut vertex
• Cut edge
• Block
• Two blocks share at most one vertex cut-vertex

Connected Components
1
3
Connected Components
1
2
4
DFS

• Depth First Search
• Use the DFS tree to find Blocks

G
DFS tree
5
Mengers Theorem (1927)

• k-connected deleting (k-1) vertices cannot
  disconnect it (V(G) gt k).
• Corollary (Dirac 1960)
• G is 2-connected ? G consists of a single block
  

6
ST-Ordering

• Numbering - v1, v2, ... , vn of V(G) so that
• (v1, vn) E(G)
• vi has a neighbor vj where j lt i
• vi also has a neighbor vk where i lt k
  (for every i , 1 lt i lt n)
• G admits an st-ordering ? G is 2-connected

Well prove it soon
7
Ear Decomposition Lemma

• Every 2-connected graph can be obtained from a
  cycle by adding paths
• For each path
• Both end points are on the current graph
• But otherwise it is disjoint from it
• Proof
• Gi G, Gi ? G the current graph.
• Pick u V(Gi) , v V(Gi) such that (u, v)
  E(G) and connect v to Gi by a shortest path.

8
G admits an st-ordering ? G is 2-connected

• G is 2-connected
• Use induction on E(G)
• Show G admits st-ordering with
• v1 u, v2, ... , vn v (u, v) E(G)
• Pick a cycle C through (u, v)
• Base
• G C
• trivial

u
v
(v5)
(v1)
v2
v4
v3
9
Continued

• C ? G

u
v
(v5)
(v1)
G
G0 C
10
Continued

• C ? G
• Add a path to it as in the ear decomposition
• Number the paths vertices so they form an
  increasing chain connecting its endpoints

u
v
(v5)
(v1)
G0 C
G1
v4
v3
Shortest path
11
Continued

• C ? G
• Add a path to it as in the ear decomposition
• Number the paths vertices so they form an
  increasing chain connecting its endpoints

u
v
(v5)
(v1)
G1
G2
v2
v4
v3
12
Pieces

• G is 2-connected
• C is a cycle in G
• Pieces
• C is Separating

P2
P1
P3
13
Pieces

• Pieces can be drawn on either side of C

P2
P1
P3
14
Pieces

• Pieces can be drawn on either side of C

P2
P1
P3
15
Pieces

• Pieces can be drawn on either side of C

P2
P1
P3
16
Pieces

• Pieces can be drawn on either side of C

P2
P1
P3
17
Pieces

• Two pieces Interlace or Conflict if they cannot
  be drawn on the same side of C without crossing
  edges
• When does this happen?

P2
G
G
a1
Cyclic order a1, b1, a3, b3
P1
a2
b1
b3
P3
a3
b2
18
Pieces

• G might be a planar graph

P2
G
P1
P3
19
Pieces

• G might be a planar graph
• If one of the conflicting pieces can be drawn on
  the other side of C

P2
G
P1
P3
20
Our Goal

• Find 2 sets (S1, S2) of pieces so that
• No two pieces in the same set conflict
• S1 S2 C G

n
n
P2
G
S1 P1, P2 S2 P3
P1
P3
21
Interlacement Graph

• Vertex set the set of pieces (with respect to
  C)
• Two vertices are connected ? the pieces interlace

Interlacement(G, C)
P2
G
P1
P3
22
Is G planar?

• G 2-connected graph with cycle C
• G is planar ?
• For each piece P (with respect to C)
• P C is a planar graph
• The Interlacement graph is bipartite
  (2-colorable)

n
23
Why?
P2
Interlacement(G, C)
G
S1
P1
S2
If Interlacement(G, C) is bipartite, we can
divide the pieces to 2 sets - like we wanted
P3
24
Why?

• Each piece combined with C is a planar graph
• And we can assemble all the pieces together so
  that they dont interlace
• So G is planar

25
Planarity Testing Recursive Algorithm

• G 2-connected graph with
• n vertices
• O(n) edges
• C is a cycle in G
• The Algorithm
• Find pieces with respect to C
• Build interlace graph
• Check if it is bipartite
• Check planarity for Cs pieces (recursively)

(
)
O(n)



O(n)
26
Questions?