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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)
Copyright Copyright 黃鈴玲黃鈴玲Ch9-Ch9-22
OutlineOutline
9.1 Properties of Planar 9.1 Properties of Planar GraphsGraphs
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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
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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 is embedded in the plane) is called a plane graph.
(b) a plane graph (c) anotherplane graph
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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.
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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
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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
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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: 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.
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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#
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Fig 9-4Fig 9-4 Two embeddings of a planar graph
(a) (b)
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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.
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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
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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
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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
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