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1. Spanning Trees Minimum Spanning Trees Kruskal’s Algorithm Example Planar Graphs Euler’s Formula Main Menu (Click on the topics below) Click here to continue Sanjay Jain, Lecturer, School of Computing

2. Spanning Trees and Planar Graphs Sanjay Jain, Lecturer, School of Computing

3. Definition: A spanning tree for a graph G is a subgraph of G that a) contains every vertex of G and b) is a tree. . . . . . . Spanning Trees

4. Definition: A spanning tree for a graph G is a subgraph of G that a) contains every vertex of G and b) is a tree. Spanning Trees . . . . . .

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6. a) Every connected finite graph G has a spanning tree. b) Any two spanning trees for a graph have the same number of edges (If G has n vertices, then spanning tree of G has n-1 edges). Proposition Proof: (of a) G is connected. Let G’=G 1. If G’ is a tree then we are done. 2. Otherwise, delete an edge from a circuit of G’ and go to 1. At the end of the above algorithm, G’ will be a spanning tree of G.

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8. . Singapore 89 . 45 New York . 34 76 KL 88 65 . . 23 Jakarta London Minimal Spanning Trees.

9. Weighted Graph. Each edge has a weight associated with it. Minimal Spanning Trees Minimal spanning tree, is a spanning tree with the minimum weight.

10. May not be unique . . 1 1 . . 1 1 Minimal Spanning Trees Minimal spanning tree can be formed by taking any three edges in the above graph.

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12. Input: Graph G, V(G), E(G), weights of edges. Output: Minimal spanning tree of G. Algorithm: 1. Initialize T to contain all vertices of G and no edges. Let E=E(G). n= number of vertices in V(G) m=0 2. While m < n-1 do 2a. Find an edge in E with least weight. 2b. Delete e from E 2c. If adding e to T does not introduce a non-trivial circuit, then add e to the edge set of T m=m+1 Endif Endwhile Kruskal’s Algorithm

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14. Example . Singapore 89 . 99 New York . 34 76 KL 88 102 . . 23 Jakarta London

15. Example . Singapore 89 . 99 New York . 34 76 KL 88 102 . . 23 Jakarta London

16. Example . Singapore 89 . 99 New York . 34 76 KL 88 102 . . 23 Jakarta London

17. Example . Singapore 89 . 99 New York . 34 76 KL 88 102 . . 23 Jakarta London

18. Example . Singapore 89 . 99 New York . 34 76 KL 88 102 . . 23 Jakarta London

19. Example . Singapore 89 . 99 New York . 34 76 KL 88 102 . . 23 Jakarta London

20. Example . Singapore 89 . 99 New York . 34 76 KL 88 102 . . 23 Jakarta London

21. Example . Singapore 89 . 99 New York . 34 76 KL 88 102 . . 23 Jakarta London

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23. Definition: A graph G is planar iff it can be drawn on a plane in such a way that edges never “cross” (I.e. edges meet only at the endpoints). . . . . . . Planar Graphs

24. A drawing of planar graph G on a plane, without any crossing, is called the plane graph representation of G . . . . . . . . Plane Graph

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26. Theorem: Suppose G is a connected simple planar graph with n3 vertices and m edges. Then, m  3n-6. Note that the above theorem is applicable only for connectedsimple graphs. Euler’s Formula

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28. Euler’s Formula Examples K5 number of vertices: 5 number of edges : 10 m  3n-6 does not hold. 10  3*5-6 =9 So K5 is not planar.

29. K4 number of vertices: 4 number of edges : 6 m  3n-6 holds. 6  3*4-6 =6 So K4 may be planar (it is actually planar as we have already seen). Euler’s Formula Examples

30. K3,3 number of vertices: 6 number of edges : 9 m  3n-6 holds. 9  3*6-6 =12 So K3,3 may be planar (however K3,3 is not planar). Euler’s Formula Examples

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32. A graph is planar iff it does not have K5 or K3,3 as a “portion” of it. There is a linear time algorithm to determine whether a given graph is planar or not. If the graph is planar, then the algorithm also gives a plane graph drawing of it. Planar Graphs

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34. Faces: The portion enclosed by edges. The outside is also a face. Let m be number of edges, n be number of vertices, and f be number of faces. Then: f = m-n+2 ----------- (1) . . . . . . . . . Proof of Euler’s Formula

35. Faces: The portion enclosed by edges. The outside is also a face. Let m be number of edges, n be number of vertices, and f be number of faces. Then: f = m-n+2 ----------- (1) Proof of Euler’s Formula . . . . . . . . .

36. Faces: The portion enclosed by edges. The outside is also a face. Let m be number of edges, n be number of vertices, and f be number of faces. Then: f = m-n+2 ----------- (1) Proof of Euler’s Formula . . . . . . . . .

37. Faces: The portion enclosed by edges. The outside is also a face. Let m be number of edges, n be number of vertices, and f be number of faces. Then: f = m-n+2 ----------- (1) Proof of Euler’s Formula . . . . . . . . .

38. Faces: The portion enclosed by edges. The outside is also a face. Let m be number of edges, n be number of vertices, and f be number of faces. Then: f = m-n+2 ----------- (1) Proof of Euler’s Formula . . . . . . . . .

39. Faces: The portion enclosed by edges. The outside is also a face. Let m be number of edges, n be number of vertices, and f be number of faces. Then: f = m-n+2 ----------- (1) Proof of Euler’s Formula . . . . . . . . 3f/2  m ----------- (2) By substituting (1) in (2) we get 3m - 3n + 6  2m or m  3n - 6 .

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