Spanning Trees Minimum Spanning Trees Kruskal’s Algorithm Example Planar Graphs Euler’s Formula

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Sanjay Jain, Lecturer, School of ComputingSanjay Jain, Lecturer, School of Computing

Spanning Trees and Planar GraphsSpanning Trees and Planar Graphs

Sanjay Jain, Lecturer, Sanjay Jain, Lecturer,

School of ComputingSchool of Computing

Spanning TreesSpanning TreesDefinition: 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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Spanning TreesSpanning TreesDefinition: 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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PropositionPropositiona) 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 Trees.

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Minimal Spanning TreesMinimal Spanning TreesWeighted Graph.Each edge has a weight associated with it.

Minimal spanning tree, is a spanning tree with the minimum weight.

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eweightTweight

Minimal Spanning TreesMinimal Spanning TreesMay not be unique

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Minimal spanning tree can be formed by taking any three edges in the above graph.

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Kruskal’s AlgorithmKruskal’s 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 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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ExampleExample

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Planar GraphsPlanar 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).

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Plane GraphPlane 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 FormulaEuler’s 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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Euler’s Formula ExamplesEuler’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.

Euler’s Formula ExamplesEuler’s Formula Examples

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 ExamplesEuler’s Formula Examples

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).

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Planar GraphsPlanar Graphs

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.

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Proof of Euler’s FormulaProof of Euler’s 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) .

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Proof of Euler’s FormulaProof of Euler’s 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) .

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...

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Proof of Euler’s FormulaProof of Euler’s 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 Euler’s FormulaProof of Euler’s 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 Euler’s FormulaProof of Euler’s 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 Euler’s FormulaProof of Euler’s 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) .

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3f/2 m ----------- (2)By substituting (1) in (2) we get3m - 3n + 6 2mor m 3n - 6

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