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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.

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.

#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); }

Cycles in the Graph are −

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

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