Print all the cycles in an undirected graph in C++

In this problem, we are given an undirected graph and we have to print all the cycles that are formed in the graph.

Undirected Graph is a graph that is connected together. All the edges of the unidirectional graph are bidirectional. It is also known as an undirected network.

Cycle in a graph data structure is a graph in which all vertices form a cycle.

Let's see an example to understand the problem better ?

Graph-

Output-

Cycle 1: 2 3 4 5
Cycle 2: 6 7 8

For this, we will make use of a few properties of the graph. You need to use graph coloring method and color all the vertices which occur in a cyclic graph. Also, if a vertex is partially visited, it will give rise to a cyclic graph. So, we will color this vertex and all next vertex till the same is reached again.

ALGORITHM

Step 1: call DFS traversal for the graph which can color the vertices.
Step 2: If a partially visited vertex is found, backtrack till the vertex is
reached again and mark all vertices in the path with a counter which is cycle number.
Step 3: After completion of traversal, iterate for cyclic edge and push them
into a separate adjacency list.
Step 4: Print the cycles number wise from the adjacency list.

Example

#include <bits/stdc++.h>
using namespace std;
const int N = 100000;
vector<int> graph[N];
vector<int> cycles[N];
void DFSCycle(int u, int p, int color[], int mark[], int par[], int& cyclenumber){
  if (color[u] == 2) {
    return;
  }
  if (color[u] == 1) {
    cyclenumber++;
    int cur = p;
    mark[cur] = cyclenumber;
    while (cur != u) {
      cur = par[cur];
      mark[cur] = cyclenumber;
    }
    return;
  }
  par[u] = p;
  color[u] = 1;
  for (int v : graph[u]) {
    if (v == par[u]) {
      continue;
    }
    DFSCycle(v, u, color, mark, par, cyclenumber);
  }
  color[u] = 2;
}
void insert(int u, int v){
  graph[u].push_back(v);
  graph[v].push_back(u);
}
void printCycles(int edges, int mark[], int& cyclenumber){
  for (int i = 1; i <= edges; i++) {
    if (mark[i] != 0)
      cycles[mark[i]].push_back(i);
  }
  for (int i = 1; i <= cyclenumber; i++) {
    cout << "Cycle " << i << ": ";
    for (int x : cycles[i])
      cout << x << " ";
    cout << endl;
  }
}
int main(){
  insert(1, 2);
  insert(2, 3);
  insert(3, 4);
  insert(4, 5);
  insert(5, 2);
  insert(5, 6);
  insert(6, 7);
  insert(7, 8);
  insert(6, 8);
  int color[N];
  int par[N];
  int mark[N];
  int cyclenumber = 0;
  cout<<"Cycles in the Graph are :\n";
  int edges = 13;
  DFSCycle(1, 0, color, mark, par, cyclenumber);
  printCycles(edges, mark, cyclenumber);
}

Output

Cycles in the Graph are ?

Cycle 1: 2 3 4 5
Cycle 2: 6 7 8