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Number of single cycle components in an undirected graph

Geek Week

Given a set of ‘n’ vertices and ‘m’ edges of an undirected simple graph (no parallel edges and no self-loop), find the number of single-cycle-components present in the graph. A single-cyclic-component is a graph of n nodes containing a single cycle through all nodes of the component.
Example: 
 

Let us consider the following graph with 15 vertices.

Input: V = 15, E = 14
       1 10  // edge 1
       1 5   // edge 2
       5 10  // edge 3
       2 9   // ..
       9 15  // ..
       2 15  // ..
       2 12  // ..
       12 15 // ..
       13 8  // ..
       6 14  // ..
       14 3  // ..
       3 7   // ..
       7 11  // edge 13
       11 6  // edge 14
Output :2
In the above-mentioned example, the two 
single-cyclic-components are composed of 
vertices (1, 10, 5) and (6, 11, 7, 3, 14) 
respectively.

Now we can easily see that a single-cycle-component is a connected component where every vertex has the degree as two. 
Therefore, in order to solve this problem we first identify all the connected components of the disconnected graph. For this, we use depth-first search algorithm. For the DFS algorithm to work, it is required to maintain an array ‘found’ to keep an account of all the vertices that have been discovered by the recursive function DFS. Once all the elements of a particular connected component are discovered (like vertices(9, 2, 15, 12) form a connected graph component ), we check if all the vertices in the component are having the degree equal to two. If yes, we increase the counter variable ‘count’ which denotes the number of single-cycle-components found in the given graph. To keep an account of the component we are presently dealing with, we may use a vector array ‘curr_graph’ as well.
 

C++




// CPP program to find single cycle components
// in a graph.
#include <bits/stdc++.h>
using namespace std;
 
const int N = 100000;
 
// degree of all the vertices
int degree[N];
 
// to keep track of all the vertices covered
// till now
bool found[N];
 
// all the vertices in a particular
// connected component of the graph
vector<int> curr_graph;
 
// adjacency list
vector<int> adj_list[N];
 
// depth-first traversal to identify all the
// nodes in a particular connected graph
// component
void DFS(int v)
{
    found[v] = true;
    curr_graph.push_back(v);
    for (int it : adj_list[v])
        if (!found[it])
            DFS(it);
}
 
// function to add an edge in the graph
void addEdge(vector<int> adj_list[N], int src,
             int dest)
{
    // for index decrement both src and dest.
    src--, dest--;
    adj_list[src].push_back(dest);
    adj_list[dest].push_back(src);
    degree[src]++;
    degree[dest]++;
}
 
int countSingleCycles(int n, int m)
{
    // count of cycle graph components
    int count = 0;
    for (int i = 0; i < n; ++i) {
        if (!found[i]) {
            curr_graph.clear();
            DFS(i);
 
            // traversing the nodes of the
            // current graph component
            int flag = 1;
            for (int v : curr_graph) {
                if (degree[v] == 2)
                    continue;
                else {
                    flag = 0;
                    break;
                }
            }
            if (flag == 1) {
                count++;
            }
        }
    }
    return(count);
}
 
int main()
{
    // n->number of vertices
    // m->number of edges
    int n = 15, m = 14;
    addEdge(adj_list, 1, 10);
    addEdge(adj_list, 1, 5);
    addEdge(adj_list, 5, 10);
    addEdge(adj_list, 2, 9);
    addEdge(adj_list, 9, 15);
    addEdge(adj_list, 2, 15);
    addEdge(adj_list, 2, 12);
    addEdge(adj_list, 12, 15);
    addEdge(adj_list, 13, 8);
    addEdge(adj_list, 6, 14);
    addEdge(adj_list, 14, 3);
    addEdge(adj_list, 3, 7);
    addEdge(adj_list, 7, 11);
    addEdge(adj_list, 11, 6);
 
    cout << countSingleCycles(n, m);
 
    return 0;
}

Java




// Java program to find single cycle components
// in a graph.
import java.util.*;
 
class GFG
{
 
static int N = 100000;
 
// degree of all the vertices
static int degree[] = new int[N];
 
// to keep track of all the vertices covered
// till now
static boolean found[] = new boolean[N];
 
// all the vertices in a particular
// connected component of the graph
static Vector<Integer> curr_graph = new Vector<Integer>();
 
// adjacency list
static Vector<Vector<Integer>> adj_list = new Vector<Vector<Integer>>();
 
// depth-first traversal to identify all the
// nodes in a particular connected graph
// component
static void DFS(int v)
{
    found[v] = true;
    curr_graph.add(v);
    for (int it = 0 ;it < adj_list.get(v).size(); it++)
        if (!found[adj_list.get(v).get(it)])
            DFS(adj_list.get(v).get(it));
}
 
// function to add an edge in the graph
static void addEdge( int src,int dest)
{
    // for index decrement both src and dest.
    src--; dest--;
    adj_list.get(src).add(dest);
    adj_list.get(dest).add(src);
    degree[src]++;
    degree[dest]++;
}
 
static int countSingleCycles(int n, int m)
{
    // count of cycle graph components
    int count = 0;
    for (int i = 0; i < n; ++i)
    {
 
        if (!found[i])
        {
            curr_graph.clear();
             
            DFS(i);
 
            // traversing the nodes of the
            // current graph component
            int flag = 1;
            for (int v = 0 ; v < curr_graph.size(); v++)
            {
                if (degree[curr_graph.get(v)] == 2)
                    continue;
                else
                {
                    flag = 0;
                    break;
                }
            }
            if (flag == 1)
            {
                count++;
            }
        }
    }
    return(count);
}
 
// Driver code
public static void main(String args[])
{
     
    for(int i = 0; i < N + 1; i++)
    adj_list.add(new Vector<Integer>());
     
    // n->number of vertices
    // m->number of edges
    int n = 15, m = 14;
    addEdge( 1, 10);
    addEdge( 1, 5);
    addEdge( 5, 10);
    addEdge( 2, 9);
    addEdge( 9, 15);
    addEdge( 2, 15);
    addEdge( 2, 12);
    addEdge( 12, 15);
    addEdge( 13, 8);
    addEdge( 6, 14);
    addEdge( 14, 3);
    addEdge( 3, 7);
    addEdge( 7, 11);
    addEdge( 11, 6);
     
 
    System.out.println(countSingleCycles(n, m));
}
}
 
// This code is contributed by Arnab Kundu

Python3




# Python3 program to find single
# cycle components in a graph.
N = 100000
 
# degree of all the vertices
degree = [0] * N
 
# to keep track of all the
# vertices covered till now
found = [None] * N
 
# All the vertices in a particular
# connected component of the graph
curr_graph = []
 
# adjacency list
adj_list = [[] for i in range(N)]
 
# depth-first traversal to identify
# all the nodes in a particular
# connected graph component
def DFS(v):
 
    found[v] = True
    curr_graph.append(v)
     
    for it in adj_list[v]:
        if not found[it]:
            DFS(it)
 
# function to add an edge in the graph
def addEdge(adj_list, src, dest):
 
    # for index decrement both src and dest.
    src, dest = src - 1, dest - 1
    adj_list[src].append(dest)
    adj_list[dest].append(src)
    degree[src] += 1
    degree[dest] += 1
 
def countSingleCycles(n, m):
 
    # count of cycle graph components
    count = 0
    for i in range(0, n):
        if not found[i]:
            curr_graph.clear()
            DFS(i)
 
            # traversing the nodes of the
            # current graph component
            flag = 1
            for v in curr_graph:
                if degree[v] == 2:
                    continue
                else:
                    flag = 0
                    break
                 
            if flag == 1:
                count += 1
     
    return count
 
# Driver Code
if __name__ == "__main__":
 
    # n->number of vertices
    # m->number of edges
    n, m = 15, 14
    addEdge(adj_list, 1, 10)
    addEdge(adj_list, 1, 5)
    addEdge(adj_list, 5, 10)
    addEdge(adj_list, 2, 9)
    addEdge(adj_list, 9, 15)
    addEdge(adj_list, 2, 15)
    addEdge(adj_list, 2, 12)
    addEdge(adj_list, 12, 15)
    addEdge(adj_list, 13, 8)
    addEdge(adj_list, 6, 14)
    addEdge(adj_list, 14, 3)
    addEdge(adj_list, 3, 7)
    addEdge(adj_list, 7, 11)
    addEdge(adj_list, 11, 6)
 
    print(countSingleCycles(n, m))
 
# This code is contributed by Rituraj Jain

C#




// C# program to find single cycle components
// in a graph.
using System;
using System.Collections.Generic;
     
class GFG
{
static int N = 100000;
 
// degree of all the vertices
static int []degree = new int[N];
 
// to keep track of all the vertices covered
// till now
static bool []found = new bool[N];
 
// all the vertices in a particular
// connected component of the graph
static List<int> curr_graph = new List<int>();
 
// adjacency list
static List<List<int>> adj_list = new List<List<int>>();
 
// depth-first traversal to identify all the
// nodes in a particular connected graph
// component
static void DFS(int v)
{
    found[v] = true;
    curr_graph.Add(v);
    for (int it = 0; it < adj_list[v].Count; it++)
        if (!found[adj_list[v][it]])
            DFS(adj_list[v][it]);
}
 
// function to add an edge in the graph
static void addEdge(int src,int dest)
{
    // for index decrement both src and dest.
    src--; dest--;
    adj_list[src].Add(dest);
    adj_list[dest].Add(src);
    degree[src]++;
    degree[dest]++;
}
 
static int countSingleCycles(int n, int m)
{
    // count of cycle graph components
    int count = 0;
    for (int i = 0; i < n; ++i)
    {
        if (!found[i])
        {
            curr_graph.Clear();
             
            DFS(i);
 
            // traversing the nodes of the
            // current graph component
            int flag = 1;
            for (int v = 0 ; v < curr_graph.Count; v++)
            {
                if (degree[curr_graph[v]] == 2)
                    continue;
                else
                {
                    flag = 0;
                    break;
                }
            }
            if (flag == 1)
            {
                count++;
            }
        }
    }
    return(count);
}
 
// Driver code
public static void Main(String []args)
{
    for(int i = 0; i < N + 1; i++)
    adj_list.Add(new List<int>());
     
    // n->number of vertices
    // m->number of edges
    int n = 15, m = 14;
    addEdge(1, 10);
    addEdge(1, 5);
    addEdge(5, 10);
    addEdge(2, 9);
    addEdge(9, 15);
    addEdge(2, 15);
    addEdge(2, 12);
    addEdge(12, 15);
    addEdge(13, 8);
    addEdge(6, 14);
    addEdge(14, 3);
    addEdge(3, 7);
    addEdge(7, 11);
    addEdge(11, 6);
     
    Console.WriteLine(countSingleCycles(n, m));
}
}
 
// This code is contributed by PrinciRaj1992

Javascript




<script>
 
// JavaScript program to find single cycle components
// in a graph.
 
let N = 100000;
 
// degree of all the vertices
let degree=new Array(N);
for(let i=0;i<N;i++)
    degree[i]=0;
// to keep track of all the vertices covered
// till now
let found=new Array(N);
for(let i=0;i<N;i++)
    found[i]=0;
 
// all the vertices in a particular
// connected component of the graph
let curr_graph = [];
 
// adjacency list
let adj_list = [];
 
// depth-first traversal to identify all the
// nodes in a particular connected graph
// component
function DFS(v)
{
    found[v] = true;
    curr_graph.push(v);
    for (let it = 0 ;it < adj_list[v].length; it++)
        if (!found[adj_list[v][it]])
            DFS(adj_list[v][it]);
}
 
// function to add an edge in the graph
function addEdge(src,dest)
{
    // for index decrement both src and dest.
    src--; dest--;
    adj_list[src].push(dest);
    adj_list[dest].push(src);
    degree[src]++;
    degree[dest]++;
}
 
function countSingleCycles(n,m)
{
    // count of cycle graph components
    let count = 0;
    for (let i = 0; i < n; ++i)
    {
   
        if (!found[i])
        {
            curr_graph=[];
               
            DFS(i);
   
            // traversing the nodes of the
            // current graph component
            let flag = 1;
            for (let v = 0 ; v < curr_graph.length; v++)
            {
                if (degree[curr_graph[v]] == 2)
                    continue;
                else
                {
                    flag = 0;
                    break;
                }
            }
            if (flag == 1)
            {
                count++;
            }
        }
    }
    return(count);
}
 
// Driver code
for(let i = 0; i < N + 1; i++)
    adj_list.push([]);
 
// n->number of vertices
// m->number of edges
let n = 15, m = 14;
addEdge( 1, 10);
addEdge( 1, 5);
addEdge( 5, 10);
addEdge( 2, 9);
addEdge( 9, 15);
addEdge( 2, 15);
addEdge( 2, 12);
addEdge( 12, 15);
addEdge( 13, 8);
addEdge( 6, 14);
addEdge( 14, 3);
addEdge( 3, 7);
addEdge( 7, 11);
addEdge( 11, 6);
 
 
document.write(countSingleCycles(n, m));
 
// This code is contributed by avanitrachhadiya2155
 
</script>
Output: 
2

 

Hence, total number of cycle graph component is found.
 

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