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Main Menu (Click on the topics below)Spanning TreesMinimum Spanning TreesKruskals AlgorithmExamplePlanar GraphsEulers Formula

Click here to continueSanjay Jain, Lecturer, School of Computing

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

Spanning TreesDefinition: A spanning tree for a graph G is a subgraph of G that a) contains every vertex of G andb) is a tree.

Spanning TreesDefinition: A spanning tree for a graph G is a subgraph of G that a) contains every vertex of G andb) is a tree.......

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Propositiona) 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=G1. 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.

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

Minimal Spanning TreesMay not be uniqueMinimal spanning tree can be formed by taking any three edges in the above graph.

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Kruskals AlgorithmInput: 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=02. While m < n-1 do2a. Find an edge in E with least weight.2b. Delete e from E2c. If adding e to T does not introduce a non-trivial circuit, then add e to the edge set of Tm=m+1EndifEndwhile

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Example....SingaporeKLJakartaLondonNew York897623883410299.

Example....SingaporeKLJakartaLondonNew York897623883410299.

Example....SingaporeKLJakartaLondonNew York897623883410299.

Example....SingaporeKLJakartaLondonNew York897623883410299.

Example....SingaporeKLJakartaLondonNew York897623883410299.

Example....SingaporeKLJakartaLondonNew York897623883410299.

Example....SingaporeKLJakartaLondonNew York897623883410299.

Example....SingaporeKLJakartaLondonNew York897623883410299.

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Planar GraphsDefinition: 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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Eulers FormulaTheorem: Suppose G is a connected simple planar graph with n3 vertices and m edges. Then, m 3n-6.

Note that the above theorem is applicable only for connected simple graphs.

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Eulers Formula ExamplesK5 number of vertices: 5 number of edges : 10 m 3n-6 does not hold.10 3*5-6 =9So K5 is not planar.

Eulers Formula ExamplesK4 number of vertices: 4 number of edges : 6 m 3n-6 holds.6 3*4-6 =6So K4 may be planar (it is actually planar as we have already seen).

Eulers Formula ExamplesK3,3 number of vertices: 6 number of edges : 9 m 3n-6 holds.9 3*6-6 =12So K3,3 may be planar (however K3,3 is not planar).

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Planar GraphsA 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.

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Proof of Eulers FormulaFaces: 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 Eulers FormulaFaces: 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 Eulers FormulaFaces: 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 Eulers FormulaFaces: 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 Eulers FormulaFaces: 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).........3f/2 m ----------- (2)By substituting (1) in (2) we get3m - 3n + 6 2mor m 3n - 6

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