1. Graph Theory Chapter 7 Eulerian Graphs 大葉大學 (Da-Yeh Univ.) 資訊工程系 (Dept. CSIE) 黃鈴玲 (Lingling Huang)

2. Copyright  黃鈴玲 Ch7-2 Outline 7.1 An Introduction to Eulerian Graphs 7.2 Characterizing Eulerian Graphs Again 7.3 The Chinese Postman Problem

3. Copyright  黃鈴玲 Ch7-3 7.1 An Introduction to Eulerian Graphs 1736, Euler solved the Königsberg Bridge Problem ( 七橋問題 ) 1736, Euler solved the Königsberg Bridge Problem ( 七橋問題 ) Q: 是否存在一 種走法，可以走 過每座橋一次， 並回到起點？

4. Copyright  黃鈴玲 Ch7-4 Königsberg Bridge Problem 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  A B C D Q: 是否存在一種走法，可以走過每條邊一次，並回 到起點？ 陸地為點 橋為邊

5. Copyright  黃鈴玲 Ch7-5 Definition: (1) An eulerian circuit of a connected multigraph is a circuit ( 點可重複、邊不可重複 ) of G that contains all the edges of G. (2) A (multi)graph with an eulerian circuit is called an eulerian (multi)graph. (3) An eulerian trail of a connected multigraph G is an open trail ( 起點終點不同的 trail) of G that contains all the edges of G.

6. Copyright  黃鈴玲 Ch7-6 u9u9 u8u8 u2u2 u3u3 u4u4 u5u5 u6u6 u7u7 u1u1 G1G1 v5v5 v4v4 v3v3 v2v2 v1v1 v6v6 G2G2 eulerian circuit: eulerian trail:    

7. Copyright  黃鈴玲 Ch7-7 Theorem 7.1: A connected multigraph G is eulerian if and only if the degree of each vertex is even. Pf: (  ) G is eulerian   eulerian circuit C  C 通過每一點時需用一條邊進入，用另一條邊離開 the degree of each vertex is even  the degree of each vertex is even ()() Suppose every vertex of G is even. (Now we construct an eulerian circuit.)

8. Copyright  黃鈴玲 Ch7-8 Choose any vertex v and begin a trail T ( 邊不 可重複 ) at v as far as possible. If w is the last vertex of T, then any edge incident with w must belong to T. Claim: w = v Pf. If w  v, then each time w is encountered on T before the last time, one edge is used to enter w and another edge is used to exit from w. Since w has even degree. There must be at least one edge incident with w that does not belong to T, a contradiction.  Since w has even degree. There must be at least one edge incident with w that does not belong to T, a contradiction.  If E(T)  E(G ), 在 G  T 中重複此法找出一個個的 circuit ，連接起來即可得 eulerian circuit.

9. Copyright  黃鈴玲 Ch7-9 v1v1 v2v2 Figure 7.4 (Algorithm 7.1, Eulerian circuit) v3v3 v5v5 v4v4 v6v6 Step 1: T 1 : v 1, v 2, v 3, v 4, v 5, v 1 Step 2: T 2 : v 3, v 5, v 6, v 3 Step 3: C = T 1  T 2 C: v 1, v 2, v 3, v 5, v 6, v 3, v 4, v 5, v 1 T2T2

10. Copyright  黃鈴玲 Ch7-10 Theorem 7.2: Let G be a nontrivial connected multigraph. Then G contains an eulerian trail if and only if G has exactly two odd vertices. Furthermore, the trail begins at one of the odd vertices and terminates at the other.

11. Copyright  黃鈴玲 Ch7-11 Homework Exercise 7.1: 1, 2

12. Copyright  黃鈴玲 Ch7-12 Outline 7.1 An Introduction to Eulerian Graphs 7.2 Characterizing Eulerian Graphs Again 7.3 The Chinese Postman Problem

13. Copyright  黃鈴玲 Ch7-13 7.2 Characterizing Eulerian Graphs Again Theorem 7.3: A connected graph G is eulerian if and only if every edge of G lies on an odd number of cycles of G.

14. Copyright  黃鈴玲 Ch7-14 Example (Figure 7.5) C 1 : u, v, x, u z x a uv y wb Consider the edge uv, it belongs to five cycles: C 2 : u, v, y, x, u C 3 : u, v, y, z, x, u C 4 : u, v, w, y, z, x, u C 5 : u, v, w, y, x, u

15. Copyright  黃鈴玲 Ch7-15 Homework Exercise 7.2: 4( a ) Ex4(a). Show that each edge of K n belongs to at least 2 n  2  1 cycles. C4:C4: Example: K 5 C3:C3: 個個 C5:C5: 個

16. Copyright  黃鈴玲 Ch7-16 Outline 7.1 An Introduction to Eulerian Graphs 7.2 Characterizing Eulerian Graphs Again 7.3 The Chinese Postman Problem

17. Copyright  黃鈴玲 Ch7-17 7.3 The Chinese Postman Problem Chinese Postman Problem: Suppose that a letter carrier must deliver mail to every house in a small town. The carrier would like to cover the route in the most efficient way and then return to the post office. Definition: For a connected graph G, an eulerian walk is a shortest closed walk covering all the edges of G.  finding an eulerian walk

18. Copyright  黃鈴玲 Ch7-18 An alternative way to solve the Chinese Postman Problem: For a given connected graph G, determine an eulerian multigraph H of minimum size that contains G as its underlying graph. e. g., 將圖形 G 中的每個 edge 都複製一份  每點 degree 都會是偶數  每點 degree 都會是偶數  新圖有 eulerian circuit 存在  新圖有 eulerian circuit 存在  the length of an eulerian walk of G is at least q but no more than 2q.  the length of an eulerian walk of G is at least q but no more than 2q.

19. Copyright  黃鈴玲 Ch7-19 Definition: A pair partition of V 0 (G) is a partition of V 0 (G) into n two-element subsets. For a pair partition , given by  ={{ u 11, u 12 }, { u 21, u 22 }, …, { u n1, u n2 }}. Let us define and let m ( G ) = min { d (  ) |  is a pair partition }. If G is not eulerian, then G contains an even number of odd vertices. Let V 0 (G) = {u 1, u 2, …, u 2n }, n  1, be the set of odd vertices of G.

20. Copyright  黃鈴玲 Ch7-20 If G is eulerian, then m(G ) = 0. Theorem 7.4 If G is a connected graph of size q, then an eulerian walk of G has length q + m ( G ). m(G ) m(G ) 代表的是 eulerian walk 中重複走的邊數 ※ How to find an eulerian walk of G ? (1) Find a pair partition  with d (  ) = m ( G ). (2) If  ={{ u 11, u 12 }, { u 21, u 22 }, …, { u n1, u n2 }}, determine shortest u i1 - u i2 paths Q i. (3) duplicate the edges of G that are on Q i. (4) An eulerian circuit in the new graph provides an eulerian walk of G.

21. Copyright  黃鈴玲 Ch7-21 How to find a pair partition  of V 0 (G) for which m ( G )= d (  )? ※ How to find a pair partition  of V 0 (G) for which m ( G )= d (  )? (1) Construct a complete weighted graph F  K 2n of order 2n, where V ( F ) = V 0 (G), the weight of an edge in F is defined as the distance between the corresponding vertices in G. (2) Determine a perfect matching of F whose weight is as small as possible. (Let m be the maximum weight of F. 將 F 中每邊的 weight w 改為 m+1  w, find a maximum matching 即可 )

22. Copyright  黃鈴玲 Ch7-22 Example (Fig 7.6, solving the Chinese Postman Problem) u1u1 u2u2 v1v1 u3u3 v2v2 v3v3 v4v4 u4u4 (1) Find odd vertices (2) Graph F : u1u1 u2u2 u4u4 u3u3 1 3 4 3 2 3 (3) Graph F’ : u1u1 u2u2 u4u4 u3u3 2 1 2 3 2 4 Max matching (4) add Q i : u2u2 v1v1 u3u3 v2v2 v3v3 v4v4 u4u4 u1u1

23. Copyright  黃鈴玲 Ch7-23 (5) Eulerian walk: u2u2 v1v1 u3u3 v2v2 v3v3 v4v4 u4u4 u1u1 u 1,e 12, u 2, e 10, v 3, e 3, v 4, e 1, u 4, e 2, v 4, e 4, v 3, e 7, v 2, e 8, u 3, e 5, v 3, e 6, u 3, e 9, v 1, e 11, u 2, e 13, u 1 e 13 e 12 e6e6 e5e5 e 11 e9e9 e8e8 e7e7 e3e3 e4e4 e2e2 e1e1 e 10

24. Copyright  黃鈴玲 Ch7-24 Homework Exercise 7.3: 1, 3 Ex1. Prove that the length of an eulerian walk for a tree of size q is 2 q.