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
Implementation:
C++
// 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;
} |
Java
// Java program to find the Dominant Set of a graph import java.util.*;
class GFG
{ static Vector<Integer> []g;
static boolean []box = new boolean [ 100000 ];
static Vector<Integer> Dominant( int ver, int edge)
{ Vector<Integer> S = new Vector<Integer>(); // set S
for ( int i = 0 ; i < ver; i++)
{
if (!box[i])
{
S.add(i);
box[i] = true ;
for ( int j = 0 ; j < ( int )g[i].size(); j++)
{
if (!box[g[i].get(j)])
{
box[g[i].get(j)] = true ;
break ;
}
}
}
}
return S;
} // Driver code public static void main(String[] args)
{ int ver, edge, x, y;
ver = 5 ; // Enter number of vertices
edge = 6 ; // Enter number of Edges
g = new Vector[ver];
for ( int i = 0 ; i < ver; i++)
g[i] = new Vector<Integer>();
// Enter all the end-points of all the Edges
// g[x--].push_back[y--] g[y--].push_back[x--]
g[ 0 ].add( 1 );
g[ 1 ].add( 0 ); // x = 1, y = 2 ;
g[ 1 ].add( 2 );
g[ 2 ].add( 1 ); // x = 2, y = 3 ;
g[ 2 ].add( 3 );
g[ 3 ].add( 2 ); // x = 3, y = 4 ;
g[ 0 ].add( 3 );
g[ 3 ].add( 0 ); // x = 1, y = 4 ;
g[ 3 ].add( 4 );
g[ 4 ].add( 3 ); // x = 4, y = 5 ;
g[ 2 ].add( 4 );
g[ 4 ].add( 2 ); // x = 3, y = 5 ;
Vector<Integer> S = Dominant(ver, edge);
System.out.print( "The Dominant Set is : { " );
for ( int i = 0 ; i < ( int )S.size(); i++)
System.out.print(S.get(i) + 1 + " " );
System.out.print( "}" );
} } // This code is contributed by Rajput-Ji |
Python3
# Python3 code to find the Dominant Set of a graph from typing import List
g = []
box = []
def Dominant(ver: int , edge: int ) - > List [ int ]:
S = [] # set S
for i in range (ver):
if not box[i]:
S.append(i)
box[i] = True
for j in range ( len (g[i])):
if not box[g[i][j]]:
box[g[i][j]] = True
break
return S
# Driver function if __name__ = = '__main__' :
ver = 5 # Enter number of vertices
edge = 6 # Enter number of Edges
for i in range (ver):
g.append([])
# Setting all index value of an array as False
box = [ False for i in range (ver)]
# Enter all the end-points of all the Edges
# g[x].append(y) g[y].append(x)
g[ 0 ].append( 1 )
g[ 1 ].append( 0 ) # x = 1, y = 2 ;
g[ 1 ].append( 2 )
g[ 2 ].append( 1 ) # x = 2, y = 3 ;
g[ 2 ].append( 3 )
g[ 3 ].append( 2 ) # x = 3, y = 4 ;
g[ 0 ].append( 3 )
g[ 3 ].append( 0 ) # x = 1, y = 4 ;
g[ 3 ].append( 4 )
g[ 4 ].append( 3 ) # x = 4, y = 5 ;
g[ 2 ].append( 4 )
g[ 4 ].append( 2 ) # x = 3, y = 5 ;
S = Dominant(ver, edge)
print ( "The Dominant Set is : {{ {} }}" . format ( " " .join( str (x + 1 ) for x in S)))
# This code is contributed by lokeshpotta20.
|
C#
// C# program to find the Dominant Set of a graph using System;
using System.Collections.Generic;
class GFG
{ static List< int > []g;
static bool []box = new bool [100000];
static List< int > Dominant( int ver, int edge)
{ List< int > S = new List< int >(); // set S
for ( int i = 0; i < ver; i++)
{
if (!box[i])
{
S.Add(i);
box[i] = true ;
for ( int j = 0; j < ( int )g[i].Count; j++)
{
if (!box[g[i][j]])
{
box[g[i][j]] = true ;
break ;
}
}
}
}
return S;
} // Driver code public static void Main(String[] args)
{ int ver, edge;
ver = 5; // Enter number of vertices
edge = 6; // Enter number of Edges
g = new List< int >[ver];
for ( int i = 0; i < ver; i++)
g[i] = new List< int >();
// Enter all the end-points of all the Edges
// g[x--].push_back[y--] g[y--].push_back[x--]
g[0].Add(1);
g[1].Add(0); // x = 1, y = 2 ;
g[1].Add(2);
g[2].Add(1); // x = 2, y = 3 ;
g[2].Add(3);
g[3].Add(2); // x = 3, y = 4 ;
g[0].Add(3);
g[3].Add(0); // x = 1, y = 4 ;
g[3].Add(4);
g[4].Add(3); // x = 4, y = 5 ;
g[2].Add(4);
g[4].Add(2); // x = 3, y = 5 ;
List< int > S = Dominant(ver, edge);
Console.Write( "The Dominant Set is : { " );
for ( int i = 0; i < ( int )S.Count; i++)
Console.Write(S[i] + 1 + " " );
Console.Write( "}" );
} } // This code is contributed by PrinciRaj1992 |
Javascript
function findDominantSet(ver, edge, g) {
let S = []; // Set S
let box = new Array(ver).fill( false );
for (let i = 0; i < ver; i++) {
if (!box[i]) {
S.push(i);
box[i] = true ;
for (let j = 0; j < g[i].length; j++) {
if (!box[g[i][j]]) {
box[g[i][j]] = true ;
break ;
}
}
}
}
return S;
} // Driver function function main() {
const ver = 5; // Enter the number of vertices
const edge = 6; // Enter the number of edges
const g = new Array(ver);
for (let i = 0; i < ver; i++) {
g[i] = [];
}
// Setting all values in the 'box' array to false
const box = new Array(100000).fill( false );
// Enter the endpoints of all the edges
g[0].push(1);
g[1].push(0); // x = 1, y = 2
g[1].push(2);
g[2].push(1); // x = 2, y = 3
g[2].push(3);
g[3].push(2); // x = 3, y = 4
g[0].push(3);
g[3].push(0); // x = 1, y = 4
g[3].push(4);
g[4].push(3); // x = 4, y = 5
g[2].push(4);
g[4].push(2); // x = 3, y = 5
const S = findDominantSet(ver, edge, g);
console.log( "The Dominant Set is : { " + S.map(val => val + 1).join( " " ) + " }" );
} main(); |
Output
The Dominant Set is : { 1 3 5 }