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.

**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

Below diagram to show 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)

**Implementation:**

Following are C++ and Java implementations of above approximate algorithm.

## C++

`// Program to print Vertex Cover of a given undirected graph ` `#include<iostream> ` `#include <list> ` `using` `namespace` `std; ` ` ` `// This class represents a undirected graph using adjacency list ` `class` `Graph ` `{ ` ` ` `int` `V; ` `// No. of vertices ` ` ` `list<` `int` `> *adj; ` `// Pointer to an array containing adjacency lists ` `public` `: ` ` ` `Graph(` `int` `V); ` `// Constructor ` ` ` `void` `addEdge(` `int` `v, ` `int` `w); ` `// function to add an edge to graph ` ` ` `void` `printVertexCover(); ` `// prints vertex cover ` `}; ` ` ` `Graph::Graph(` `int` `V) ` `{ ` ` ` `this` `->V = V; ` ` ` `adj = ` `new` `list<` `int` `>[V]; ` `} ` ` ` `void` `Graph::addEdge(` `int` `v, ` `int` `w) ` `{ ` ` ` `adj[v].push_back(w); ` `// Add w to v’s list. ` ` ` `adj[w].push_back(v); ` `// Since the graph is undirected ` `} ` ` ` `// The function to print vertex cover ` `void` `Graph::printVertexCover() ` `{ ` ` ` `// Initialize all vertices as not visited. ` ` ` `bool` `visited[V]; ` ` ` `for` `(` `int` `i=0; i<V; i++) ` ` ` `visited[i] = ` `false` `; ` ` ` ` ` `list<` `int` `>::iterator 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. ` ` ` `for` `(i= adj[u].begin(); i != adj[u].end(); ++i) ` ` ` `{ ` ` ` `int` `v = *i; ` ` ` `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` `i=0; i<V; i++) ` ` ` `if` `(visited[i]) ` ` ` `cout << i << ` `" "` `; ` `} ` ` ` `// Driver program to test methods of graph class ` `int` `main() ` `{ ` ` ` `// Create a graph given in the above diagram ` ` ` `Graph g(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(); ` ` ` ` ` `return` `0; ` `} ` |

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## Java

`// 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(); ` ` ` `} ` `} ` `// This code is contributed by Aakash Hasija ` |

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## Python3

`# Python3 program to print Vertex Cover ` `# of a given undirected graph ` `from` `collections ` `import` `defaultdict ` ` ` `# This class represents a directed graph ` `# using adjacency list representation ` `class` `Graph: ` ` ` ` ` `def` `__init__(` `self` `, vertices): ` ` ` ` ` `# No. of vertices ` ` ` `self` `.V ` `=` `vertices ` ` ` ` ` `# Default dictionary to store graph ` ` ` `self` `.graph ` `=` `defaultdict(` `list` `) ` ` ` ` ` `# Function to add an edge to graph ` ` ` `def` `addEdge(` `self` `, u, v): ` ` ` `self` `.graph[u].append(v) ` ` ` ` ` `# The function to print vertex cover ` ` ` `def` `printVertexCover(` `self` `): ` ` ` ` ` `# Initialize all vertices as not visited. ` ` ` `visited ` `=` `[` `False` `] ` `*` `(` `self` `.V) ` ` ` ` ` `# Consider all edges one by one ` ` ` `for` `u ` `in` `range` `(` `self` `.V): ` ` ` ` ` `# An edge is only picked when ` ` ` `# both visited[u] and visited[v] ` ` ` `# are false ` ` ` `if` `not` `visited[u]: ` ` ` ` ` `# 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. ` ` ` `for` `v ` `in` `self` `.graph[u]: ` ` ` `if` `not` `visited[v]: ` ` ` ` ` `# 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` `j ` `in` `range` `(` `self` `.V): ` ` ` `if` `visited[j]: ` ` ` `print` `(j, end ` `=` `' '` `) ` ` ` ` ` `print` `() ` ` ` `# Driver code ` ` ` `# Create a graph given in ` `# the above diagram ` `g ` `=` `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() ` ` ` `# This code is contributed by Prateek Gupta ` |

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Output:

0 1 3 4 5 6

Time Complexity of above algorithm is O(V + E).

**Exact Algorithms:**

Although the problem is NP complete, it can be solved in polynomial time for following types of graphs.

1) Bipartite Graph

2) Tree Graph

The problem to check whether there is a vertex cover of size smaller than or equal to a given number k can also be solved in polynomial time if k is bounded by O(LogV) (Refer this)

We will soon be discussing exact algorithms for vertex cover.

This article is contributed by **Shubham**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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