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# 图论的介绍 - PowerPoint PPT Presentation

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### 实际生活中的图论Graph Model

Pspice

Social Network

High School Dating

corporate e-mail

Reference: Bearman, Moody and Stovel, 2004

image by Mark Newman

Protein interaction network

Reference: Jeong et al, Nature Review | Genetics

The Internet

The Internet as mapped by The Opte Project

http://www.opte.org

More Applications
• Hypertexts
• 网页包含到其他网页的超链接。整个Web是一个图. 搜索引擎需要图处理算法。
• Matching
• 职位招聘，如何有效将职位与应聘者匹配？
• Schedules
• 工程项目的任务安排，如何满足限制条件，并在最短时间内完成？
• Program structure
• 大型软件系统，函数（模块）之间调用关系。编译器分析调用关系图确定如何最好分配资源才能使程序更有效率。

### Graph Problems and Algorithms

• 可以求出從某一点到图上其他任一点的最短路径

Some graph-processing problems
• Path. Is there a path between s to t?
• Shortest path. What is the shortest path between s and t?
• Longest path. What is the longest simple path between s and t?
• Cycle. Is there a cycle in the graph?
• Euler tour. Is there a cycle that uses each edge exactly once?
• Hamilton tour. Is there a cycle that uses each vertex exactly once?
• Connectivity. Is there a way to connect all of the vertices?
• MST. What is the best way to connect all of the vertices?
• Biconnectivity. Is there a vertex whose removal disconnects the graph?
• Planarity. Can you draw the graph in the plane with no crossing edges?

First challenge: Which of these problems is easy? difficult? intractable?

### 图论的术语

Set of vertices connected pairwise by edges.

V: 顶点的集合

E: 边的集合

V= {a,b,c,d,e}

E= {(a,b),(a,c),(a,d),

(b,e),(c,d),(c,e),

(d,e)}

(directed acyclic graph)

### 一些特殊的图

• 任意两点之间都有一条边与其相连的图称为完全图，以Kn來表示，n为顶点数

• A graph that can be decomposed into two partite sets but not fewer is bipartite
• It is a complete bipartite if its vertices can be divided into two non-empty groups, A and B. Each vertex in A is connected to B, and vice-versa

The graph is bipartite

Complete bipartite graph K2,3

• A planar graph is a graph that can be embedded in a plane so that no edges intersect

Planar:

=

NOT Planar:

• 8个顶点(V=8)
• 12条边(E=12)
• 6个面 (F=6)
• 8-12+6=2

• 树(tree):连通无简单回路无向图
• 若有n个顶点，則有n-1条边
• 任两点之间仅有一条路径
• 生成树(spanning tree):包括图中所有的顶点，并且是一棵树

tree

A connected graph G

Spanning trees of G