A STUDY ON SOME NAMED GRAPHS TO FIND THE MINIMUM SPANNING TREE (MST) USING GREEDY ALGORITHMS

In this paper, Greedy Algorithms such as Kruskal’s, Prim’s, Boruvka’s, Reverse-delete Algorithm were applied on different types of graphs like Mobius-Kantor graph, Durer graph, Golomb graph to find the (MST).


Definition
If there is a path [1] between each 3. Mobius-Kantor Graph:

Mobius-Kantor Graph using Kruskal's Algorithm
Step 1: Arrange the edges in ascending sequence Step 2: Pick an edge with the smallest w i.e.) connect an edge AB and LC having weight 1.
between each pair of vertices then the graph is said to be Connected  Step 3: Pick an edge with the next smallest weight that does not form a cycle.
i.e.) connect an edge BC, BK, DM and HI having weight 2.
Step 4: Repeat step 2, until all the vertices are connect

Mobius-Kantor Graph using Prim's Algorithm
Step 1: Select any vertex and connect that vertex i.e.) connect an edge AB having weight 1.
Step 2: Now treat the vertices A and B as i.e) connect an edge BK or BC having weight 2. Now we connect an edge BK.

Mobius-Kantor graph using Boruvka's Algorithm
Step 1: Select the minimum weighted edge in every vertex. Step 2: Select the weighted edge that connects the graph and that does not form a cycle. Step 3: Repeat step 2, until we obtain a MST.

Mobius-Kantor graph using Reverse-Delete Algorithm
Step 1: Arrange the edges in descending sequence.  Step 2: Delete the edges with the highest weight and check if deletion does not lead to disconnection. i.e.) delete an edge PK and LO having weight 10.

Durer Graph using Kruskal's Algorithm
Step 1: Arrange the edges in ascending sequence. Step 2: Pick an edge with the smallest wei i.e.) connect an edge AB and HJ having   Step 3: Pick an edge with the next smallest weight that does not form a cycle.
i.e.) connect an edge BC and IK having weight 2.

Durer Graph using Prim's Algorithm
Step 1: Select any vertex and connect that vertex to the edge having smallest weight.
i.e.) connect an edge AB having weight 1.

Durer graph using Boruvka's Algorithm
Step 1: Select the minimum weighted edge in every vertex.  Step 2: Select the weighted edge that connects the graph and make sure a cycle is not formed. Step 3: Repeat step 3, until we obtain a MST.

Durer graph using Reverse-Delete Algorithm
Step 1: Arrange the edges in descending sequence.  Step 2: Delete the edges with the highest weight and check if deletion does not lead to disconnection. i.e.) delete an edge GK having weight 11.

Golomb Graph using Kruskal's Algorithm
Step 1: Arrange the edges in ascending sequence. Step 2: Pick an edge with the smallest weight that does not form a cycle.
i.e.) connect an edge AB, CH and FG having weight 1.

Figure 5.1(a)
Step 3: Pick an edge with the next smallest weight and make sure a cycle is not formed.
i.e.) connect an edge BC having weight 2. Step 4: Repeat step 2, until all the vertices are connected and a MST is obtained.

Golomb Graph using Prim's Algorithm
Step 1: Select any vertex and connect that vertex to the edge having smallest weight.
i.e.) connect an edge AB having weight 1. Step 3: Repeat step 2, as far as all the vertices are connected and a MST is obtained.

Golomb graph using Boruvka's Algorithm
Step 1: Select the minimum weighted edge in every vertex. Step 2: Select the weighted edge that connects the Step 3: Repeat step 2, as far as w

Golomb graph using Reverse-Delete Algorithm
Step 1: Arrange the edges in descending sequence. Step 2: Delete the edges with the highest weight and check if deletion does not lead to disconnection. i.e.) delete an edge EF having weight 10.

Figure 5.4(a)
Step 3: Repeat step 2, until we obtain a MST.

Remark:
Till now there is no MST for Mobius-Kantor graph, Durer graph and Golomb graph using Greedy Algorithms. In this paper, we explain about the Greedy Algorithms to find the MST for these named graphs.

Conclusion:
In this paper, we discussed about the Greedy Algorithms such as Kruskal's, Prim's, Boruvka's and Reverse-Delete Algorithm to find the MST for some named graphs. Among these Kruskal's Algorithm is the best to find the MST.