大葉大學 資訊工程系 黃鈴玲

 G. Agnarsson and R. Greenlaw, Graph Theory: Modeling, Applications, and Algorithms, Pearson,  G. Chartrand and O. R. Oellermann, Applied and Algorithmic Graph Theory, McGraw-Hill,

 Ch1 - Introduction to Graph Theory  Ch2 - Basic Concepts in Graph Theory  Ch3 - Trees and Forests  Ch4 - Spanning Trees  Ch5 - Fundamental Properties of Graphs and Digraphs  Ch6 - Connectivity and Flow  Ch7 - Planar Graphs  Ch8 - Graph Coloring  Ch10 - Independence, Dominance, and Matchings  Ch12 - Graph Algorithms 3

 Problem 1.1: The Bridges of Kőnigsberg 5 Problem: Make a round trip through downtown Kőnigsberg, traversing each bridge exactly once.

6 Ans: 因為每次經過一個點，都需要從一條邊進入該點，再用另 一條邊離開，所以經過每個點一次要使用掉一對邊。  每個點上連接的邊數必須是偶數才行  此種走法不存在 (Chapter 5) Q: 是否存在一種走法，可以走過每條邊一次，並回到起點？ B1B1 B2B2 I1I1 I2I2

 Problem 1.2: World Wide Web Communities 7 旅行社網頁 航空公司網站 Complete Bipartite Graph 網頁連結

8  Problem 1.3: Job Assignments Jobs: Applicants: qualified Bipartite Graph Problem 1.8: Is the company able to meet its hiring need? If so, provide a possible set of hires that meet their needs. Ans: No (Ch10 Matching)

9  Problem 1.4: Storing Volatile Chemicals Problem: C 1, C 2, …, C 7 為有揮發性的化 學藥品，有些不能存放在一起， (An edge between C i and C j indicates a grave danger in storing these chemicals in the same warehouse.) 至少需幾個倉庫？ Ans: 4 (Ch8 Graph Coloring)

 Set, element, empty set, subset, union, intersection, disjoint, difference (A\B), cardinality (|A|, 即 A 集合的元素個數 )  symmetric difference of A and B: A  B = (A\B)  (B\A)  power set of S: P(S) = { all subsets of S}  k-tuple: (a 1, a 2, …, a k )  Cartesian product of A 1, A 2, …, A k is A 1  A 2  …  A k ={ (a 1, a 2, …, a k ) | a i  A i for each i } 10

 A graph or a general graph is an ordered triple G = (V, E,  ), where 1. V  . 2. V  E = . 3.  : E  P(V) is a map such that |  (e)|  {1, 2} for each e  E.  Vertex ( 點 ): element of V (V 也常寫成 V(G))  Edge ( 邊 ): element of E (E 也常寫成 E(G))   : edgemap   (e): endvertices ( 兩端點 ) of the edge e  (Note: V and E can be infinite.) 11

12 G=(V, E,  ) V={u 1, u 2, u 3, u 4, u 5 } E={e 1, e 2, e 3, e 4, e 5, e 6 }  (e 1 )={u 1, u 2 }  (e 2 )=  (e 3 )={u 1, u 3 }  (e 4 )={u 2, u 3 }  (e 5 )={u 3, u 4 }  (e 6 )={u 4 } u 5 is called isolated.

13  u, v : vertices of a graph G  u is called an endvertex ( 端點 ) of e.  u and v are adjacent (or neighbors)  u and e are incident. (adjacent 用在點與點連接，以及邊與邊連接， incident 用在點與邊連接 )  loop:  u v e Multiple edges, parallel edges:

 Simple graph: a graph having no multiple edges or any loop.  can be omitted  G=(V, E) 14

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19 Exercise 11, 12

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21 f u or hbors is the N(u 1 ) = {u 2, u 3 } N[u 1 ] = {u 1, u 2, u 3 } N(u 4 ) = N[u 4 ] = {u 3, u 4 } Exercise 17

22 pf. 在計算 degree 總和時，每條邊會被計算兩次， 所以 degree 的總和等於邊數的兩倍。 degree sum = 12 E(G) = 6

23 pf. If the number of vertices with odd degree is odd, then the degree sum must be odd.  (degree 是奇數的點，一定會有偶數個 ) The null graph N n is 0-regular. The cycle C n is 2-regular. The complete graph K n is (n  1)-regular. The complete (m,n)-bipartite graph K m,n is a regular graph if and only if m=n. Every k-regular graph on n vertices has kn/2 edges. Exercise 13

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25 is a subgraph of G if We write G ‘  G.

26 If W={w 1, w 2, …, w m }, we write G[w 1, w 2, …, w m ] instead of G[{w 1, w 2, …, w m }].

27 Exercise 14, 15, 19

28 u v e e' (  唸成 eta)

29 G = (V, E,  ) V = {u 1, u 2, u 3, u 4, u 5 } E = {e 1, e 2, e 3, e 4, e 5, e 6 }

30 ( 有向圖去掉邊的方向性後，所得之無向圖 )

31 Simple digraph: a digraph without directed loops and parallel directed edges.

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33 N + (u 3 ) ={u 1, u 4 } N  (u 3 ) ={u 1, u 2 } and the

34 ( 所有點 indegree 總和 = outdegree 總和 = 邊數 ) A directed cycle is balanced and regular. Exercise: 補充： Draw a nonregular balanced digraph of 5 vertices. 26