Dominant Set of a Graph

In graph theory, a dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is adjacent to at least one member of D. The domination number is the number of vertices in a smallest dominating set for G. 

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Examples:

Input : A graph with 4 vertex and 4 edges 
Output :  The Dominant Set S= { a, b } or { a, d } or { a, c } and more.

Input : A graph with 6 vertex and 7 edges 
Output : The Dominant Set S= { a, d, f } or { e, c } and more.

It is believed that there may be no efficient algorithm that finds a smallest dominating set for all graphs, but there are efficient approximation algorithms. 

Algorithm :

• First we have to initialize a set 'S' as empty
• Take any edge 'e' of the graph connecting the vertices ( say A and B )
• Add one vertex between A and B ( let say A ) to our set S
• Delete all the edges in the graph connected to A
• Go back to step 2 and repeat, if some edge is still left in the graph
• The final set S is a Dominant Set of the graph

Implementation:

The Dominant Set is : { 1 3 5 }