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.


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
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// C++ program to find the Dominant Set of a graph
#include <bits/stdc++.h>
using namespace std;
  
vector<vector<int> > g;
bool box[100000];
  
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[0].push_back(1);
    g[1].push_back(0); // x = 1, y = 2 ;
    g[1].push_back(2);
    g[2].push_back(1); // x = 2, y = 3 ;
    g[2].push_back(3);
    g[3].push_back(2); // x = 3, y = 4 ;
    g[0].push_back(3);
    g[3].push_back(0); // x = 1, y = 4 ;
    g[3].push_back(4);
    g[4].push_back(3); // x = 4, y = 5 ;
    g[2].push_back(4);
    g[4].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;
}

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Output:

The Dominant Set is : { 1 3 5 }

Reference : wiki



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