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Graph Theory

Chapter 9Planar Graphs


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Outline

9.1 Properties of Planar Graphs


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9.1 Properties of Planar

GraphsDefinition:

A graph that can be drawn in the planewithout any of its edges intersectingis calleda planar graph. A graph that is so drawn inthe plane is also said to be embedded (orimbedded) in the plane.

Applications:(1) circuit layout problems(2) Three house and three utilities problem


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Fig 9-1

(a) planar,not a plane graph

Definition:

A planar graph G that is drawn in the plane

so that no two edges intersect (that is, G isembedded in the plane)is called a plane graph.

(b) a plane graph (c) anotherplane graph


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Definition:

Let G be a plane graph. The connected piecesof the plane that remain when the vertices andedges ofG are removed are called the regionsofG.

Note.A given planar graph can give rise toseveral different plane graph.

Fig 9-2

R3: exteriorG1

R1

R2G1has 3 regions.


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Definition:

Every plane graph has exactly one unboundedregion, called the exterior region. The verticesand edges ofG that are incident with a regionRform a subgraph ofG called the boundary ofR.

G2

G2has only 1 region.

Fig 9-2


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Fig 9-2

R1

R2

R3

R4

R5

G3

v1 v2

v3

v4

v5

v6 v7

v8 v9

G3has 5 regions.

Boundary ofR1:v1 v2

v3

Boundary ofR5:v

1

v2

v3

v4v5

v6 v7

v9


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Observe:(1)Each cycle edge belongs to the boundary

of two regions.(2) Each bridge is on the boundary of only one region.

(exterior)


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pf: (by induction on q)

Thm 9.1: (Eulers Formula)IfG is a connected plane graph with p

vertices, q edges, andr regions, thenp q +r = 2.

(basis) Ifq = 0, then GK1; sop = 1,r =1,andp q +r = 2.

(inductive) Assume the result is true for any

graph with q =k - 1 edges, wherek 1.

Let G be a graph withk edges. Suppose G hasp vertices andr regions.


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IfG is a tree, then G hasp vertices,p-1

edges and 1 region.p q +r =p(p-1) + 1 = 2.

IfG is not a tree, then some edge e ofG is on a

cycle.Hence G-e is a connected plane graph havingorderp and sizek-1, andr-1 regions.

p k 1) + (r 1) = 2 (by assumption)

p k +r = 2#


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Fig 9-4Two embeddings of a planar graph

(a) (b)


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Definition:

A plane graph G is called maximal planar if, forevery pair u, v of nonadjacent vertices ofG, thegraph G+uv is nonplanar.

Thus, in any embedding of a maximal planar graph Gof order at least 3, the boundary of every region ofG

is a triangle.


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pf:

Thm 9.2: IfG is a maximal planar graphwithp 3 vertices and q edges, then

q = 3p 6.

Embed the graph G in the plane, resulting inr regions.

Since the boundary of every region of G is atriangle, every edge lies on the boundary oftwo regions.

qrR

R23|}ofboundarytheofedgesthe{|

region

p q +r = 2.

p q + 2q/3 = 2.

q = 3p

-6


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pf:

Cor. 9.2(a): IfG is a maximal planarbipartite graph withp 3 vertices and qedges, then q = 2p 4.

The boundary of every region is a 4-cycle.

Cor. 9.2(b): IfG is a planar graph withp 3vertices and q edges, then

q 3p 6.

pf: If G is not maximal planar, we can add edgesto G to produce a maximal planar graph.

By Thm. 9.2 .

4r = 2q p q + q/2 = 2 q = 2p 4.


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pf:

Thm 9.3: Every planar graph contains avertex of degree 5 or less.

Let G be a planar graph ofp vertices andq edges.

Ifdeg(v)6 for every vV(G)

2q 6p

)(

6)deg(

GVv

pv


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Fig 9-5Two important nonplanar graph

K5K3,3


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pf:

Thm 9.4: The graphsK5 andK3,3 arenonplanar.

(1) K5 hasp= 5 vertices and q = 10 edges.

(2) SupposeK3,3 is planar, and consider any

embedding ofK3,3 in the plane.

q > 3p 6 K5 is nonplanar.

Suppose the embedding hasr regions.

p q +r = 2 r = 5

K3,3is bipartiteThe boundary of every region has 4 edges.

182|}ofboundarytheofedgesthe{|4region

qRrR


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Definition:

Asubdivision of a graph G is a graph obtainedby inserting vertices (of degree 2) into theedges ofG.

p. 31 Gsubdivisiongraph G


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Fig 9-6Subdivisions of graphs.

G

H

F

His a subdivision ofG.

F is not a subdivision ofG.


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Fig 9-7The Petersen graph is nonplanar.

(a) Petersen

Thm 9.5: (Kuratowskis Theorem)A graph is planar if and only if it contains no

subgraph that is isomorphic to or is asubdivision ofK5 orK3,3.

(b) Subdivision ofK3,3

1 2 3

654

1

2 3

4 5 6

7

8 9

10


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Homework

Exercise 9.1:1, 2, 3, 5

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