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# Cycles in an Undirected Graph

Last modified: November 25, 2022

## 1. Overview

In this tutorial, we’re going to learn to **detect cycles in an undirected graph using Depth-First Search (DFS).**

## 2. What Is a Graph?

A graph is a data structure that comprises a restricted set of vertices (or nodes) and a set of edges that connect these vertices.

We can define a graph , with a set of vertices , and a set of edges . Every edge connects two vertices, and we can show it as , where and are connected vertices.

For example, if there is an edge between two vertices _{ }and , then we call them associated. We can then also call these two as adjacent (neighbor) vertices.

Mathematically, we can show a graph ( vertices, edges) as:

We can categorize graphs into two groups:

First, if edges can only be traversed in one direction, we call the graph *directed*.

For example, if a directed edge connects vertex 1 and 2, we can traverse from vertex 1 to vertex 2, but the opposite direction (from 2 to 1) is not allowed. So, we can say that is not equal to .

But, if the edges are bidirectional, we call the graph *undirected*.

For example, if an undirected edge connects vertex 1 and 2, we can traverse from vertex 1 to vertex 2 and from 2 to 1. We can then say that is equal to .

## 3. What Is a Cycle?

**In graph theory, a path that starts from a given vertex and ends at the same vertex is called a cycle.**

Cycle detection is a major area of research in computer science. The complexity of detecting a cycle in an undirected graph is .

In the example below, we can see that nodes 3-4-5-6-3 result in a cycle:

## 4. Cycle Detection

Next, then, let’s learn how to detect cycles in an undirected graph. Specifically, let’s use DFS to do it.

As a quick reminder, DFS places vertices into a stack. We start with some vertex and push it onto the stack. Then:

- If has unvisited neighbors, we push one of those neighbors, , onto the stack and repeat this step (now is )
- Once has no unvisited neighbors, we pop it off of the stack and return to step 1

Now, **to detect a cycle, we can adjust DFS’s logic a bit:** If has a visited neighbor that:

- is equal to , or
- is not equal to ‘s parent

**then we’ve got a cycle.**

Putting this into pseudocode, we get:

And now we can use it to detect cycles in undirected graphs by calling .

## 5. Conclusion

In this quick tutorial, we explored how to detect cycles in undirected graphs – basing our algorithm on Depth-First Search.