Planar Graphs
Mathematics for Computer Science
MIT 6.042J/18.062J
Planar Graphs
Albert R Meyer,
March 19, 2010
Albert R Meyer,
lec 7F.1
March 19, 2010
lec 7F.2
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Planar Graphs
4 Continuous Faces here
A graph is planar if there
is a way to draw it in the
plane without edges crossing.
3
2
4
1
the outside face
continuous face
::=connected region
Albert R Meyer,
March 19, 2010
Albert R Meyer,
lec 7F.3
Region Boundaries
abca
Albert R Meyer,
a
c
d
March 19, 2010
lec 7F.4
Region Boundaries
b
a
March 19, 2010
c
d
lec 7F.5
Albert R Meyer,
b
March 19, 2010
lec 7F.6
1
Region Boundaries
Region Boundaries
abca
a
abca
b
abda
a
c
d
Albert R Meyer,
March 19, 2010
Albert R Meyer,
lec 7F.7
acda
b
bcdb
d
Albert R Meyer,
a
c
rstur
w
v
March 19, 2010
e
efge
g
abcefgec da
Albert R Meyer,
March 19, 2010
lec 7F.13
Region Boundaries: Dongle
s
r
t
y
x
w
u
Albert R Meyer,
c
abcda
lec 7F.10
s
x
lec 7F.9
f
d
March 19, 2010
y
March 19, 2010
b
Region Boundaries: Dongle
r
c
Region Boundaries: Bridge
abca
abda
bcdb
d
Region Boundaries
a
abda
b
u
lec 7F.15
Albert R Meyer,
v
t
stvxy xv wvt urs
March 19, 2010
lec 7F.16
2
Planar Embedding
Same graph, different embeddings
A planar embedding is a
connected graph along with
its face boundary cycles
(same graph may have
different embeddings)
Albert R Meyer,
March 19, 2010
2 length 5 faces
Albert R Meyer,
lec 7F.17
Recursive Def: Planar Embeddings
March 19, 2010
Albert R Meyer,
lec 7F.19
Constructor: Split a Face
a
March 19, 2010
t
z
x
a
y
b
y
awxbyza → awxba, abyza
lec 7F.21
axyza
u
b
z
March 19, 2010
lec 7F.20
Constructor: Add a Bridge
w
Albert R Meyer,
lec 7F.18
Two constructor cases:
1)  Add edge across a face
(splits face in two)
2) Add bridge between
connected components
(merges 2 outer faces)
face
Albert R Meyer,
March 19, 2010
Adding an edge to an embedding
Base: a graph consisting of
•  single vertex, v,
•  with a single face:
length 0 cycle from v to v,
is a PE.
v
v
graph
length 3 face
length 7 face
w
x
v
btuvwb
Albert R Meyer,
March 19, 2010
lec 7F.22
3
Constructor: Add a Bridge
Constructor: Add a Bridge
t
z
a
y
u
b
v
btuvwb
axyza,
Albert R Meyer,
March 19, 2010
lec 7F.23
Team Problem
March 19, 2010
w
x
v
axyza, btuvwb → axyzabtuvw ba
Albert R Meyer,
March 19, 2010
lec 7F.24
v–e+f=2
lec 7F.25
Albert R Meyer,
March 19, 2010
lec 7F.26
Adding an edge to a drawing
Proof by structural
induction on embeddings:
base case: 1 vertex
Constructor case (split face):
v stays the same
e increases by 1
f increases by 1
so v – e + f stays the same
v = 1, e = 0, f = 1
1–0+1=2
March 19, 2010
b
If a planar embedding has
v vertices, e edges, and f
faces, then
Euler's Formula
Albert R Meyer,
a
u
Euler's Formula
Problem 1
Albert R Meyer,
z
y
w
x
t
lec 7F.27
Albert R Meyer,
March 19, 2010
lec 7F.28
4
Adding an edge to a drawing
Planar Properties
Constructor case (add bridge):
v = v1 + v 2
- e = -(e1 + e2 + 1)
f = f1 + f2 – 1 (two outer faces
2= 2 + 2-2
Albert R Meyer,
merge into one)
March 19, 2010
lec 7F.29
e ≤ 3v-6
Cor: K5 is not planar
pf: v = 5, e = 10
10  3·5 - 6
March 19, 2010
lec 7F.31
Planar Properties
March 19, 2010
lec 7F.30
Planar Properties
e ≤ 3v-6
Cor: Every planar graph has
a vertex of degree ≤ 5
pf: suppose all degrees ≥ 6
Then
6v ≤ ∑ degrees = 2e ≤ 6v-12
contradiction!
Albert R Meyer,
March 19, 2010
lec 7F.32
Cor: There
are at most
5 regular polyhedra
(proof in Notes)
a vertex of degree ≤ 5
Therefore,
every planar graph
is 6-colorable
March 19, 2010
Albert R Meyer,
Euler's Formula
Cor: Every planar graph has
Albert R Meyer,
so3(e-v+2)
3f ≤
so
total face=length
= 2e
combining with Euler
e ≤ 3v-6
Planar Properties
Albert R Meyer,
•  an edge appears twice on faces
•  face length ≥ 3
(for v ≥ 3)
lec 7F.33
Albert R Meyer,
March 19, 2010
lec 7F.34
5
Team Problems
Problems
2&3
Albert R Meyer,
March 19, 2010
lec 7F.35
6
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Spring 2010
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