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1北京邮电大学计算机学院--离散数学-10.7-planargraph.ppt
* * College of Computer Science Technology, BUPT Homework 设G是连通平面图,G的每个面的度数大于等于l (l ? 3),则 e ? l(v-2)/(l-2) 用上述性质证明K3,3不是平面图。 §10.7 6, 8, 12, 18, 24, 30 College of Computer Science Technology, BUPT Yang Juan yangjuan@bupt.edu.cn College of Computer Science Technology Beijing University of Posts Telecommunications Discrete Mathematical Structures Planar Graphs * * College of Computer Science Technology, BUPT Planar Graphs – 平面图 A graph is called planar if it can be drawn in the plane in such a way that no two edges cross. Example of a planar graph: The clique on 4 nodes. * * College of Computer Science Technology, BUPT Is K5 planar? * * College of Computer Science Technology, BUPT What about K3,3 ? * * College of Computer Science Technology, BUPT Why Planar? The problem of drawing a graph in the plane arises frequently in VLSI layout problems. * * College of Computer Science Technology, BUPT Regions, faces – 面 A planar representation of a graph splits the plane into regions that we call faces, including an unbounded region. one face two faces * * College of Computer Science Technology, BUPT Question Can you redraw this graph as a planar graph so as to alter the number of its faces? * * College of Computer Science Technology, BUPT Example This graph has 6 vertices 8 edges and 4 faces vertices – edges + faces = 2 * * College of Computer Science Technology, BUPT Example This graph has 7 vertices 12 edges and 7 faces vertices – edges + faces = 2 * * College of Computer Science Technology, BUPT Euler Theorem 1752 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 of G. Then r = e ? v +2. * * College of Computer Science Technology, BUPT Proof: By induction on the # of cycles of G. Base case: G has no cycles. G is connected so it must be a tree. Thus, e = v - 1 and r = 1. * * College of Computer Science Technology, BUPT Inductive step Suppose G has at least one
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