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Spanning Tree

A subgraph T of a connected graph G is called spanning tree of G if T is a tree and T include all vertices of G.

Discrete Mathematics Minimum Spanning Tree

Minimum Spanning Tree:

Suppose G is a connected weight graph i.e., each edge of G is assigned a non-negative number called the weight of edge, then any spanning tree T of G is assigned a total weight obtained by adding the weight of the edge in T.

A minimum spanning tree of G is a tree whose total weight is as small as possible.

Kruskal's Algorithm to find a minimum spanning tree: This algorithm finds the minimum spanning tree T of the given connected weighted graph G.

  1. Input the given connected weighted graph G with n vertices whose minimum spanning tree T, we want to find.
  2. Order all the edges of the graph G according to increasing weights.
  3. Initialize T with all vertices but do include an edge.
  4. Add each of the graphs G in T which does not form a cycle until n-1 edges are added.

Example1: Determine the minimum spanning tree of the weighted graph shown in fig:

Discrete Mathematics Minimum Spanning Tree

Solution: Using kruskal's algorithm arrange all the edges of the weighted graph in increasing order and initialize spanning tree T with all the six vertices of G. Now start adding the edges of G in T which do not form a cycle and having minimum weights until five edges are not added as there are six vertices.

Edges       Weights       Added or Not
(B, E)           2                   Added
(C, D)           3                   Added
(A, D)           4                   Added
(C, F)           4                   Added
(B, C)           5                   Added
(E, F)           5                   Not added
(A, B)           6                   Not added
(D, E)           6                   Not added
(A, F)           7                   Not added

Step1:

Discrete Mathematics Minimum Spanning Tree

Step2:

Discrete Mathematics Minimum Spanning Tree

Step3:

Discrete Mathematics Minimum Spanning Tree

Step4:

Discrete Mathematics Minimum Spanning Tree

Step5:

Discrete Mathematics Minimum Spanning Tree

Step6: Edge (A, B), (D, E) and (E, F) are discarded because they will form the cycle in a graph.

So, the minimum spanning tree form in step 5 is output, and the total cost is 18.

Discrete Mathematics Minimum Spanning Tree

Example2: Find all the spanning tree of graph G and find which is the minimal spanning tree of G shown in fig:

Discrete Mathematics Minimum Spanning Tree

Solution: There are total three spanning trees of the graph G which are shown in fig:

Discrete Mathematics Minimum Spanning Tree
Discrete Mathematics Minimum Spanning Tree
Discrete Mathematics Minimum Spanning Tree

To find the minimum spanning tree, use the KRUSKAL'S ALGORITHM. The minimal spanning tree is shown in fig:

Edges       Weights       Added or Not
(E, F)           1                   Added
(A, B)           2                   Added
(C, D)           2                   Added
(B, C)           3                   Added
(D, E)           3                   Added
(B, D)           6                   Not Added

Discrete Mathematics Minimum Spanning Tree

The first one is the minimum spanning having the minimum weight = 11.


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