usaco-guide/content/5_Gold/Cyc.md

---
id: cyc
title: Cycle Detection
author: Siyong Huang
prerequisites: 
 - 
     - Silver - Functional Graphs
 - 
     - Gold - Breadth First Search
description: A simple cycle is a non-empty path of distinct edges that start and end at the same vertex such that no vertex appears more than once. Describes how to detect cycles in both directed and undirected graphs.
---

*Cycle detection* determines properties of cycles in a directed or undirected graph, such as whether each node of the graph is part of a cycle or just checking whether a cycle exists. 

## Undirected Graphs

 - [CSES Round Trip](https://cses.fi/problemset/task/1669)

BFS-Cycle

## Directed Graphs

 - [CSES Round Trip II](https://cses.fi/problemset/task/1678)

The same general idea is implemented below to find any cycle in a directed graph (if one exists).

```cpp
//UNTESTED

bool visited[MAXN], on_stack[MAXN];
vector<int> adj[MAXN];
vector<int> cycle;
bool dfs(int n)
{
	visited[n] = on_stack[n] = true;
	for(int u:adj[n])
	{
		if(on_stack[u])
			return cycle.push_back(v), cycle.push_back(u), on_stack[n] = on_stack[u] = false, true;
		else if(!visited[u])
		{
			if(dfs(u))
				if(on_stack[n])
					return cycle.push_back(n), on_stack[n] = false, true;
				else
					return false;
			if(!cycle.empty())
				return false;
		}
	}
	on_stack[n] = false;
	return false;
}
int main()
{
	//take input, etc
	for(int i = 1;cycle.empty() && i <= N;i++)
		dfs(i);
	if(cycle.empty())
		printf("No cycle found!\n");
	else
	{
		reverse(cycle.begin(), cycle.end());
		printf("Cycle of length %u found!\n", cycle.size());
		for(int n : cycle) printf("%d ", n);
		printf("\n");
	}
}
```

### Problems

 - [CSES Graph Girth](https://cses.fi/problemset/task/1707)
   - shortest cycle