graph theory chapter 9 planar graphs

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Graph Theory Graph Theory Chapter 9 Chapter 9 Planar Graphs Planar Graphs 大大大大 大大大大 (Da-Yeh Univ.) (Da-Yeh Univ.) 大大大大大 大大大大大 (Dept. CSIE) (Dept. CSIE) 大大大 大大大 (Lingling Huang) (Lingling Huang)

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Graph Theory Chapter 9 Planar Graphs. 大葉大學 (Da-Yeh Univ.) 資訊工程系 (Dept. CSIE) 黃鈴玲 (Lingling Huang). Outline. 9.1 Properties of Planar Graphs. 9.1 Properties of Planar Graphs. Definition: - PowerPoint PPT Presentation

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Page 1: Graph Theory Chapter 9 Planar Graphs

Graph TheoryGraph Theory

Chapter 9Chapter 9 Planar GraphsPlanar Graphs

大葉大學大葉大學 (Da-Yeh (Da-Yeh Univ.)Univ.)資訊工程系資訊工程系 (Dept. (Dept. CSIE)CSIE)黃鈴玲黃鈴玲 (Lingling (Lingling Huang)Huang)

Page 2: Graph Theory Chapter 9 Planar Graphs

Copyright Copyright 黃鈴玲黃鈴玲Ch9-Ch9-22

OutlineOutline

9.1 Properties of Planar 9.1 Properties of Planar GraphsGraphs

Page 3: Graph Theory Chapter 9 Planar Graphs

Copyright Copyright 黃鈴玲黃鈴玲Ch9-Ch9-33

9.1 Properties of 9.1 Properties of Planar GraphsPlanar Graphs

Definition: A graph that can be drawn in the plane

without any of its edges intersecting is called a planar graph. A graph that is so drawn in the plane is also said to be embedded (or imbedded) in the plane.

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

Page 4: Graph Theory Chapter 9 Planar Graphs

Copyright Copyright 黃鈴玲黃鈴玲Ch9-Ch9-44

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 is embedded in the plane) is called a plane graph.

(b) a plane graph (c) anotherplane graph

Page 5: Graph Theory Chapter 9 Planar Graphs

Copyright Copyright 黃鈴玲黃鈴玲Ch9-Ch9-55

Definition:Definition: Let G be a plane graph. The connected

pieces of the plane that remain when the vertices and edges of G are removed are called the regions of G.

Note. A given planar graph can give rise to several different plane graph.

Fig 9-2Fig 9-2

R3: exteriorG1

R1

R2G1 has 3 regions.

Page 6: Graph Theory Chapter 9 Planar Graphs

Copyright Copyright 黃鈴玲黃鈴玲Ch9-Ch9-66

Definition: Every plane graph has exactly one

unbounded region, called the exterior region. The vertices and edges of G that are incident with a region R form a subgraph of G called the boundary of R.

G2

G2 has only 1 region.

Fig 9-2Fig 9-2

Page 7: Graph Theory Chapter 9 Planar Graphs

Copyright Copyright 黃鈴玲黃鈴玲Ch9-Ch9-77

Fig 9-2Fig 9-2

R1

R2

R3

R4

R5

G3 v1 v2

v3

v4v5

v6 v7

v8 v9

G3 has 5 regions.

Boundary of R1:v1 v2

v3

Boundary of R5:v1 v2

v3

v4v5

v6 v7

v9

Page 8: Graph Theory Chapter 9 Planar Graphs

Copyright Copyright 黃鈴玲黃鈴玲Ch9-Ch9-88

Observe:(1) Each cycle edge belongs to the boundary of two regions.(2) Each bridge is on the boundary of only one region.

(exterior)

Page 9: Graph Theory Chapter 9 Planar Graphs

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pf: pf: (by induction on (by induction on qq))

Thm 9.1: (Euler’s Formula)Thm 9.1: (Euler’s Formula) If If GG is a connected plane graph with is a connected plane graph with pp vertices, vertices, qq edges, and edges, and rr regions, then regions, then pp qq + + rr = = 22..

(basis) If q = 0, then G K1; so p = 1, r =1, and pp qq + + rr = 2 = 2.

(inductive) Assume the result is true for anygraph with q = k 1 edges, where k 1.

Let G be a graph with k edges. Suppose G hasp vertices and r regions.

Page 10: Graph Theory Chapter 9 Planar Graphs

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If G is a tree, then G has p vertices, p1 edges and 1 region.

pp qq + + rr = = pp – ( – (pp1) + 1 = 21) + 1 = 2.

If G is not a tree, then some edge e of G is on a cycle.

Hence Ge is a connected plane graph having order p and size k1, and r1 regions.

pp kk1)1) + ( + (rr1)1) = 2 = 2 (by assumption)(by assumption)

pp kk + + rr = 2 = 2#

Page 11: Graph Theory Chapter 9 Planar Graphs

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Fig 9-4Fig 9-4 Two embeddings of a planar graph

(a) (b)

Page 12: Graph Theory Chapter 9 Planar Graphs

Copyright Copyright 黃鈴玲黃鈴玲Ch9-Ch9-1212

Definition: A plane graph G is called maximal planar

if, for every pair u, v of nonadjacent vertices of G, the graph G+uv is nonplanar.

Thus, in any embedding of a maximal planar graph G of order at least 3, the boundary of every region of Gis a triangle.

Page 13: Graph Theory Chapter 9 Planar Graphs

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pf:pf:

Thm 9.2: Thm 9.2: If If GG is a maximal planar is a maximal planar graph with graph with pp 3 vertices and 3 vertices and qq edges, edges, thenthen qq = 3 = 3pp 6. 6.

Embed the graph G in the plane, resulting in r regions.Since the boundary of every region of G is atriangle, every edge lies on the boundary oftwo regions.

qrRR

23|} ofboundary theof edges the{|region

pp qq + + rr = 2. = 2.

pp qq + + 22qq // 33 = 2. = 2.

qq = 3 = 3pp 6 6

Page 14: Graph Theory Chapter 9 Planar Graphs

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pf:pf:

Cor. 9.2(a): Cor. 9.2(a): If If GG is a maximal planar is a maximal planar bipartite graph with bipartite graph with pp 3 vertices and 3 vertices and qq edges, then edges, then qq = = 22pp 4 4..

The boundary of every region is a 4-cycle.

Cor. 9.2(b): Cor. 9.2(b): If If GG is a planar graph with is a planar graph with pp 3 vertices and 3 vertices and qq edges, then edges, then qq 3 3pp 6. 6.

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

By Thm. 9.2 得證 .

4r = 2q pp qq + + qq // 22 = 2 = 2 qq = = 22pp 4. 4.

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pf:pf:

Thm 9.3: Thm 9.3: Every planar graph contains Every planar graph contains a vertex of degree 5 or lessa vertex of degree 5 or less..

Let G be a planar graph of pp vertices and vertices andqq edges edges.

If deg(v) 6 for every vV(G)

2q 6p

)(

6)deg(GVv

pv

Page 16: Graph Theory Chapter 9 Planar Graphs

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Fig 9-5Fig 9-5 Two important nonplanar graph

K5K3,3

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pf:pf:

Thm 9.4: Thm 9.4: The graphs The graphs KK55 and and KK3,33,3 are are nonplanarnonplanar..

(1)(1) K K55 has pp = 5 vertices and qq = = 1010 edges edges.

(2) Suppose KK3,33,3 is planar, and consider any embedding of KK3,33,3 in the plane.

q > 3p 6 KK55 is nonplanar. is nonplanar.

Suppose thethe embedding has r regions. pp qq + + rr = = 2 2 r = 5

KK3,33,3 is bipartite The boundary of every region has 4 edges. 182|} ofboundary theof edges the{|4

region

qRrR

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Definition: A subdivision of a graph G is a graph

obtained by inserting vertices (of degree 2) into the edges of G.

注意:此定義與 p. 31 中定義 G 的 subdivision graph 為在 G 的每條邊上各加一點並不相同。

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Fig 9-6Fig 9-6 Subdivisions of graphs.

GH

F

H is a subdivision of G.

F is not a subdivision of G.

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Fig 9-7Fig 9-7 The Petersen graph is nonplanar.

(a) Petersen

Thm 9.5: (Kuratowski’s Theorem)Thm 9.5: (Kuratowski’s Theorem)A graph is planar if and only if it A graph is planar if and only if it contains no subgraph that is isomorphic contains no subgraph that is isomorphic to or is a subdivision of to or is a subdivision of KK55 or or KK3,33,3..

(b) Subdivision of K3,3

1 2 3

654

1

2 3

4 5 6

7

8 9

10

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HomeworkHomework

Exercise 9.1:Exercise 9.1: 1, 2, 3, 5 1, 2, 3, 5