# 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. 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.
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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

 `// C++ program to find the Dominant Set of a graph ` `#include ` `using` `namespace` `std; ` ` `  `vector > g; ` `bool` `box; ` ` `  `vector<``int``> Dominant(``int` `ver, ``int` `edge) ` `{ ` `    ``vector<``int``> S; ``// set S ` `    ``for` `(``int` `i = 0; i < ver; i++) { ` `        ``if` `(!box[i]) { ` `            ``S.push_back(i); ` `            ``box[i] = ``true``; ` `            ``for` `(``int` `j = 0; j < (``int``)g[i].size(); j++) { ` `                ``if` `(!box[g[i][j]]) { ` `                    ``box[g[i][j]] = ``true``; ` `                    ``break``; ` `                ``} ` `            ``} ` `        ``} ` `    ``} ` `    ``return` `S; ` `} ` ` `  `// Driver function ` `int` `main() ` `{ ` `    ``int` `ver, edge, x, y; ` ` `  `    ``ver = 5; ``// Enter number of vertices ` `    ``edge = 6; ``// Enter number of Edges ` `    ``g.resize(ver); ` ` `  `    ``// Setting all index value of an array as 0 ` `    ``memset``(box, 0, ``sizeof``(box));  ` ` `  `    ``// Enter all the end-points of all the Edges ` `    ``// g[x--].push_back[y--]      g[y--].push_back[x--] ` `    ``g.push_back(1); ` `    ``g.push_back(0); ``// x = 1, y = 2 ; ` `    ``g.push_back(2); ` `    ``g.push_back(1); ``// x = 2, y = 3 ; ` `    ``g.push_back(3); ` `    ``g.push_back(2); ``// x = 3, y = 4 ; ` `    ``g.push_back(3); ` `    ``g.push_back(0); ``// x = 1, y = 4 ; ` `    ``g.push_back(4); ` `    ``g.push_back(3); ``// x = 4, y = 5 ; ` `    ``g.push_back(4); ` `    ``g.push_back(2); ``// x = 3, y = 5 ; ` ` `  `    ``vector<``int``> S = Dominant(ver, edge); ` `    ``cout << ``"The Dominant Set is : { "``; ` `    ``for` `(``int` `i = 0; i < (``int``)S.size(); i++) ` `        ``cout << S[i] + 1 << ``" "``; ` `    ``cout << ``"}"``; ` `    ``return` `0; ` `} `

Output:

```The Dominant Set is : { 1 3 5 }
```

Reference : wiki

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