# cycles

In graph theory, a cycle represents a path within the graph where only the starting and ending nodes are equal. Furthermore, cycles can be double-connected links between neighboring nodes or self-loops. The cycles detection algorithm implemented within MAGE works on an undirected graph and has no guarantee of node order in the output. The implemented algorithm (Gibb) is described in the 1982 MIT report called "Algorithmic approaches to circuit enumeration problems and applications (opens in a new tab)" (Boon Chai Lee, 1982). The problem is not solvable in polynomial time. It is based on finding all subsets of fundamental cycles which takes about $\mathcal{O}(2^(|E|-|V|+1))$ time where $E$ represents a set of relationships (edges) and $V$ represents a set of nodes (vertices) of the given graph.

TraitValue
Module typealgorithm
ImplementationC++
Graph directionundirected
Edge weightsunweighted
Parallelismsequential

## Procedures

### get()

#### Output:

• cycle_id: integer ➡ Incremental cycle ID of a certain node. There is no guarantee on how nodes are going to be ordered within cycle. The cycle can be represented with a minimum of one ID, where it stands for self-loop.
• node: Vertex ➡ Vertex object with all properties which is going to be related to the cycle ID it belongs to.

#### Usage:

CALL cycles.get()
YIELD cycle_id, node;

## Example

### Database state

The database contains the following data: Created with the following Cypher queries:

MERGE (a:Node {id: 0}) MERGE (b:Node {id: 1}) CREATE (a)-[:RELATION]->(b);
MERGE (a:Node {id: 0}) MERGE (b:Node {id: 2}) CREATE (a)-[:RELATION]->(b);
MERGE (a:Node {id: 1}) MERGE (b:Node {id: 2}) CREATE (a)-[:RELATION]->(b);
MERGE (a:Node {id: 2}) MERGE (b:Node {id: 3}) CREATE (a)-[:RELATION]->(b);
MERGE (a:Node {id: 2}) MERGE (b:Node {id: 4}) CREATE (a)-[:RELATION]->(b);
MERGE (a:Node {id: 3}) MERGE (b:Node {id: 4}) CREATE (a)-[:RELATION]->(b);

### Identify cycles

Get the values using the following query:

CALL cycles.get()
YIELD cycle_id, node
RETURN cycle_id, node;

Results:

+-----------------+-----------------+
| cycle_id        | node            |
+-----------------+-----------------+
| 0               | (:Node {id: 2}) |
| 0               | (:Node {id: 0}) |
| 0               | (:Node {id: 1}) |
| 1               | (:Node {id: 4}) |
| 1               | (:Node {id: 2}) |
| 1               | (:Node {id: 3}) |
+-----------------+-----------------+