# Print all the cycles in an undirected graph

Given an undirected graph, print all the vertices that form cycles in it.

**Pre-requisite:** Detect Cycle in a directed graph using colors

In the above diagram, the cycles have been marked with dark green color. The output for the above will be

1st cycle:3 5 4 6

2nd cycle:11 12 13

**Approach: **Using the graph coloring method, mark all the vertex of the different cycles with unique numbers. Once the graph traversal is completed, push all the similar marked numbers to an adjacency list and print the adjacency list accordingly. Given below is the algorithm:

- Insert the edges into an adjacency list.
- Call the DFS function which uses the coloring method to mark the vertex.
- Whenever there is a partially visited vertex,
**backtrack**till the current vertex is reached and mark all of them with cycle numbers. Once all the vertexes are marked, increase the cycle number. - Once Dfs is completed, iterate for the edges and push the same marked number edges to another adjacency list.
- Iterate in the another adjacency list and print the vertex cycle-number wise.

Below is the implementation of the above approach:

`// C++ program to print all the cycles ` `// in an undirected graph ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` `const` `int` `N = 100000; ` ` ` `// variables to be used ` `// in both functions ` `vector<` `int` `> graph[N]; ` `vector<` `int` `> cycles[N]; ` ` ` `// Function to mark the vertex with ` `// different colors for different cycles ` `void` `dfs_cycle(` `int` `u, ` `int` `p, ` `int` `color[], ` ` ` `int` `mark[], ` `int` `par[], ` `int` `& cyclenumber) ` `{ ` ` ` ` ` `// already (completely) visited vertex. ` ` ` `if` `(color[u] == 2) { ` ` ` `return` `; ` ` ` `} ` ` ` ` ` `// seen vertex, but was not completely visited -> cycle detected. ` ` ` `// backtrack based on parents to find the complete cycle. ` ` ` `if` `(color[u] == 1) { ` ` ` ` ` `cyclenumber++; ` ` ` `int` `cur = p; ` ` ` `mark[cur] = cyclenumber; ` ` ` ` ` `// backtrack the vertex which are ` ` ` `// in the current cycle thats found ` ` ` `while` `(cur != u) { ` ` ` `cur = par[cur]; ` ` ` `mark[cur] = cyclenumber; ` ` ` `} ` ` ` `return` `; ` ` ` `} ` ` ` `par[u] = p; ` ` ` ` ` `// partially visited. ` ` ` `color[u] = 1; ` ` ` ` ` `// simple dfs on graph ` ` ` `for` `(` `int` `v : graph[u]) { ` ` ` ` ` `// if it has not been visited previously ` ` ` `if` `(v == par[u]) { ` ` ` `continue` `; ` ` ` `} ` ` ` `dfs_cycle(v, u, color, mark, par, cyclenumber); ` ` ` `} ` ` ` ` ` `// completely visited. ` ` ` `color[u] = 2; ` `} ` ` ` `// add the edges to the graph ` `void` `addEdge(` `int` `u, ` `int` `v) ` `{ ` ` ` `graph[u].push_back(v); ` ` ` `graph[v].push_back(u); ` `} ` ` ` `// Function to print the cycles ` `void` `printCycles(` `int` `edges, ` `int` `mark[], ` `int` `& cyclenumber) ` `{ ` ` ` ` ` `// push the edges that into the ` ` ` `// cycle adjacency list ` ` ` `for` `(` `int` `i = 1; i <= edges; i++) { ` ` ` `if` `(mark[i] != 0) ` ` ` `cycles[mark[i]].push_back(i); ` ` ` `} ` ` ` ` ` `// print all the vertex with same cycle ` ` ` `for` `(` `int` `i = 1; i <= cyclenumber; i++) { ` ` ` `// Print the i-th cycle ` ` ` `cout << ` `"Cycle Number "` `<< i << ` `": "` `; ` ` ` `for` `(` `int` `x : cycles[i]) ` ` ` `cout << x << ` `" "` `; ` ` ` `cout << endl; ` ` ` `} ` `} ` ` ` `// Driver Code ` `int` `main() ` `{ ` ` ` ` ` `// add edges ` ` ` `addEdge(1, 2); ` ` ` `addEdge(2, 3); ` ` ` `addEdge(3, 4); ` ` ` `addEdge(4, 6); ` ` ` `addEdge(4, 7); ` ` ` `addEdge(5, 6); ` ` ` `addEdge(3, 5); ` ` ` `addEdge(7, 8); ` ` ` `addEdge(6, 10); ` ` ` `addEdge(5, 9); ` ` ` `addEdge(10, 11); ` ` ` `addEdge(11, 12); ` ` ` `addEdge(11, 13); ` ` ` `addEdge(12, 13); ` ` ` ` ` `// arrays required to color the ` ` ` `// graph, store the parent of node ` ` ` `int` `color[N]; ` ` ` `int` `par[N]; ` ` ` ` ` `// mark with unique numbers ` ` ` `int` `mark[N]; ` ` ` ` ` `// store the numbers of cycle ` ` ` `int` `cyclenumber = 0; ` ` ` `int` `edges = 13; ` ` ` ` ` `// call DFS to mark the cycles ` ` ` `dfs_cycle(1, 0, color, mark, par, cyclenumber); ` ` ` ` ` `// function to print the cycles ` ` ` `printCycles(edges, mark, cyclenumber); ` `} ` |

*chevron_right*

*filter_none*

**Output: **

Cycle Number 1: 3 4 5 6 Cycle Number 2: 11 12 13

**Time Complexity:** O(N + M), where N is number of vertex and M is the number of edges.

**Auxiliary Space:** O(N + M)

## Recommended Posts:

- Cycles of length n in an undirected and connected graph
- Product of lengths of all cycles in an undirected graph
- Convert the undirected graph into directed graph such that there is no path of length greater than 1
- Program to find the diameter, cycles and edges of a Wheel Graph
- Clone an Undirected Graph
- Connected Components in an undirected graph
- Sum of degrees of all nodes of a undirected graph
- Eulerian Path in undirected graph
- Detect cycle in an undirected graph
- Find k-cores of an undirected graph
- Number of Triangles in an Undirected Graph
- Detect cycle in an undirected graph using BFS
- Count number of edges in an undirected graph
- Find if an undirected graph contains an independent set of a given size
- Check if there is a cycle with odd weight sum in an undirected graph

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.