Cycle

Definition

A cycle is a path that returns to its starting node.

Example:

A -> B -> C -> A

The path starts at A and comes back to A.

Cycle = closed path.

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Cycle In An Undirected Graph

Example:

A -- B
B -- C
C -- A

These edges form a cycle:

A -- B -- C -- A

You can start at A, move through connected nodes, and return to A.

An undirected cycle closes back on itself.

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Cycle In A Directed Graph

In a directed graph, the arrows must form a closed route.

Example:

A -> B
B -> C
C -> A

This is a directed cycle.

But:

A -> B
B -> C
A -> C

is not a directed cycle, because there is no edge returning to A.

Directed cycles must follow arrows back to the start.

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Self-Loop As Cycle

A self-loop can be treated as a cycle of length 1.

Example:

A -> A

The path starts at A and immediately returns to A.

A self-loop is the smallest cycle when allowed by the graph.

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Why Cycles Matter

Cycles indicate repetition or circular dependency.

Examples:

A depends on B
B depends on C
C depends on A

This creates a circular structure.

In computation, cycles can also represent repeated steps.

Cycles show when structure can loop back on itself.

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Final Summary

• A cycle is a path that returns to its starting node.
• Undirected cycles close through connected edges.
• Directed cycles must follow arrows back to the start.
• A self-loop can be a cycle of length 1.
• Cycles often represent repetition or circular dependency.