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

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

Spanning Trees 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 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

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Proposition 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). 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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Minimal Spanning Trees Singapore KL Jakarta London New York

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

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

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Kruskal’s Algorithm 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

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Example.... Singapore KL Jakarta London New York

Example.... Singapore KL Jakarta London New York

Example.... Singapore KL Jakarta London New York

Example.... Singapore KL Jakarta London New York

Example.... Singapore KL Jakarta London New York

Example.... Singapore KL Jakarta London New York

Example.... Singapore KL Jakarta London New York

Example.... Singapore KL Jakarta London New York

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Planar Graphs 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)

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

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Euler’s Formula 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 connected simple graphs.

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Euler’s Formula Examples K5K5 number of vertices: 5 number of edges : 10 m  3n-6 does not hold. 10  3*5-6 =9 So K 5 is not planar.

Euler’s Formula Examples K4K4 number of vertices: 4 number of edges : 6 m  3n-6 holds. 6  3*4-6 =6 So K 4 may be planar (it is actually planar as we have already seen).

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

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Planar Graphs A graph is planar iff it does not have K 5 or K 3,3 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.

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Proof of Euler’s Formula 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 (1)

Proof of Euler’s Formula 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 (1)

Proof of Euler’s Formula 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 (1)

Proof of Euler’s Formula 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 (1)

Proof of Euler’s Formula 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 (1)

Proof of Euler’s Formula 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 (1) f/2  m (2) By substituting (1) in (2) we get 3m - 3n + 6  2m or m  3n - 6

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