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1、1,8.7 可平面图 Planar Graphs,2,8.7 可平面图 Planar Graphs,例:,有六个结点的图如上, 试问:能否转变成与其等价的,但没有任何相交线的平面上的图? 结论:不能,3,8.7 可平面图 Planar Graphs,DEFINITION A graph is called planar(可平面的) if it can be drawn in the plane without any edges crossing. Such a drawing is called a planar representation(平面表示) of the graph,4,例:,
2、8.7 可平面图 Planar Graphs,5,例:,8.7 可平面图 Planar Graphs,6,8.7 可平面图 Planar Graphs,一个图的可平面表示把平面分割成一些面,包括一个无界的面。包围每个面的边界的长度称为面的次数,记为Deg(R)。,7,8.7 可平面图 Planar Graphs,EULERS FORMULA Let G be a connected planar simple graph with e edges and v vertices. Let r be the number of regions in a planar representation
3、of G. Then r=e-v+2.,8,证明:用数学归纳法 归纳基础: 面数r=1,r=e-v+2成立。 面数r=2,G为一多边形,且e=v=3(e=v=4), 得e-v+2=3-3+2=r成立, 或e-v+2=4-4+2=r成立;,8.7 可平面图 Planar Graphs,9,归纳步骤: 设图G的面为r时,r=e-v+2成立。 证明面数为r=r+1时,等式也成立。 (a)先构成图G,其中点数为v,边数为e,面数为r; (b)在G中,加入一条长度为L的简单通路(L1),且与G共有二个结点,从而使G变为G;,8.7 可平面图 Planar Graphs,10,(c) e-v+2 =(e+
4、L)-(v+(L-1)+2 =e+L-v-L+1+2 =e-v+2+1 =r+1 =r 定理成立,8.7 可平面图 Planar Graphs,L条边 (L-1)个点,G,11,8.7 可平面图 Planar Graphs,COROLLARY If G is a connected planar simple graph with e edges and v vertices, where v3, then e3v-6。,12,8.7 可平面图 Planar Graphs,证明: G为简单连通平面图 每一面至少用三条或更多条边构成, =所有面的边的总数。 因此边的总数 3r( 包含重复计算的边
5、),13,8.7 可平面图 Planar Graphs,一条边是在至多二个面的边界中, 各面的实际总边数一定有3r( 2e) 即 成立,14,8.7 可平面图 Planar Graphs,由欧拉定理:,15,8.7 可平面图 Planar Graphs,例:证明K5图不是平面图 K5图中,v=5,e=10, 3*5-6=910 K5图不为平面图 思考:证明K3,3不是平面图,16,8.7 可平面图 Planar Graphs,COROLLARY If G is a connected planar simple graph, then G has a vertex of degree not
6、exceeding five. COROLLARY If a connected planar simple graph has e edges and v vertices with v3 and no circuits of length three, then e2v-4.,17,8.7 可平面图 Planar Graphs,Elementary subdivision(初等细分) Removing an edge u, v and adding a new vertex w together with edges u, w and w, v The graphs G1 and G2 a
7、re called homeomorphic(同胚) if they can be obtained from the same graph by a sequence of elementary subdivisions.,18,8.7 可平面图 Planar Graphs,例:同胚图,19,8.7 可平面图 Planar Graphs,THEOREM A graph is nonplanar if and only if it contains a subgraph homeomorphic to K3,3 or K5.,20,8.7 可平面图 Planar Graphs,Example: Determine whether the following graph is planar,21,8.7 可平面图 Planar Graphs,Example:,22,8.7 可平面图 Planar Graphs,Example:,