A vertex cover of an undirected graph is a subset of its vertices such that for every edge (u, v) of the graph, either ‘u’ or ‘v’ is in vertex cover. Although the name is Vertex Cover, the set covers all edges of the given graph. Given an undirected graph, the vertex cover problem is to find minimum size vertex cover.

Following are some examples.

Vertex Cover Problem is a known NP Complete problem, i.e., there is no polynomial time solution for this unless P = NP. There are approximate polynomial time algorithms to solve the problem though. Following is a simple approximate algorithm adapted from CLRS book.

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Approximate Algorithm for Vertex Cover:

1) Initialize the result as {}
2) Consider a set of all edges in given graph.  Let the set be E.
3) Do following while E is not empty
...a) Pick an arbitrary edge (u, v) from set E and add 'u' and 'v' to result
...b) Remove all edges from E which are either incident on u or v.
4) Return result 
Following diagram taken from CLRS book shows execution of above approximate algorithm.

How well the above algorithm perform?
It can be proved that the above approximate algorithm never finds a vertex cover whose size is more than twice the size of minimum possible vertex cover (Refer this for proof)

Vertex Cover Problem is a known NP Complete problem, i.e., there is no polynomial time solution for this unless P = NP. There are approximate polynomial time algorithms to solve the problem though. Following is a simple approximate algorithm adapted from CLRS book.

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Approximate Algorithm for Vertex Cover:

1) Initialize the result as {}
2) Consider a set of all edges in given graph.  Let the set be E.
3) Do following while E is not empty
...a) Pick an arbitrary edge (u, v) from set E and add 'u' and 'v' to result
...b) Remove all edges from E which are either incident on u or v.
4) Return result

Following diagram taken from CLRS book shows execution of above approximate algorithm.

How well the above algorithm perform?
It can be proved that the above approximate algorithm never finds a vertex cover whose size is more than twice the size of minimum possible vertex cover (Refer this for proof)

// Java Program to print Vertex Cover of a given undirected graph
import java.io.*;
import java.util.*;
import java.util.LinkedList;
 
// This class represents an undirected graph using adjacency list
class Graph
{
    private int V;   // No. of vertices
 
    // Array  of lists for Adjacency List Representation
    private LinkedList<Integer> adj[];
 
    // Constructor
    Graph(int v)
    {
        V = v;
        adj = new LinkedList[v];
        for (int i=0; i<v; ++i)
            adj[i] = new LinkedList();
    }
 
    //Function to add an edge into the graph
    void addEdge(int v, int w)
    {
        adj[v].add(w);  // Add w to v's list.
        adj[w].add(v);  //Graph is undirected
    }
 
    // The function to print vertex cover
    void printVertexCover()
    {
        // Initialize all vertices as not visited.
        boolean visited[] = new boolean[V];
        for (int i=0; i<V; i++)
            visited[i] = false;
 
        Iterator<Integer> i;
 
        // Consider all edges one by one
        for (int u=0; u<V; u++)
        {
            // An edge is only picked when both visited[u]
            // and visited[v] are false
            if (visited[u] == false)
            {
                // Go through all adjacents of u and pick the
                // first not yet visited vertex (We are basically
                //  picking an edge (u, v) from remaining edges.
                i = adj[u].iterator();
                while (i.hasNext())
                {
                    int v = i.next();
                    if (visited[v] == false)
                    {
                         // Add the vertices (u, v) to the result
                         // set. We make the vertex u and v visited
                         // so that all edges from/to them would
                         // be ignored
                         visited[v] = true;
                         visited[u]  = true;
                         break;
                    }
                }
            }
        }
 
        // Print the vertex cover
        for (int j=0; j<V; j++)
            if (visited[j])
              System.out.print(j+" ");
    }
 
    // Driver method
    public static void main(String args[])
    {
        // Create a graph given in the above diagram
        Graph g = new Graph(7);
        g.addEdge(0, 1);
        g.addEdge(0, 2);
        g.addEdge(1, 3);
        g.addEdge(3, 4);
        g.addEdge(4, 5);
        g.addEdge(5, 6);
 
        g.printVertexCover();
    }
}
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