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Cycle algorithms detect and enumerate cycles in graphs. A cycle is a path that starts and ends at the same node.

​
isAcyclic

function isAcyclic(graph: Graph): boolean
Checks whether the graph contains no cycles. Complexity: O(V + E) When to use:
  • Validating directed acyclic graphs (DAGs)
  • Checking for circular dependencies
  • Ensuring topological sorting is possible
  • Validating tree structures
Algorithm:
  • Directed graphs: DFS with color marking (white/gray/black)
  • Undirected graphs: DFS with parent tracking

​
Example

import { createGraph, isAcyclic } from '@statelyai/graph';

const dag = createGraph({
  nodes: [{ id: 'a' }, { id: 'b' }],
  edges: [{ id: 'ab', sourceId: 'a', targetId: 'b' }],
});

isAcyclic(dag); // true

​
Detecting circular dependencies

const modules = createGraph({
  nodes: [
    { id: 'utils.ts' },
    { id: 'api.ts' },
    { id: 'auth.ts' },
  ],
  edges: [
    { id: '1', sourceId: 'utils.ts', targetId: 'api.ts' },
    { id: '2', sourceId: 'api.ts', targetId: 'auth.ts' },
    { id: '3', sourceId: 'auth.ts', targetId: 'utils.ts' }, // cycle!
  ],
});

if (!isAcyclic(modules)) {
  console.error('Circular dependency detected!');
}

​
Validating before topological sort

import { isAcyclic, getTopologicalSort } from '@statelyai/graph';

if (isAcyclic(graph)) {
  const sorted = getTopologicalSort(graph);
  console.log('Valid ordering:', sorted?.map(n => n.id));
} else {
  console.error('Cannot sort - graph contains cycles');
}

​
getCycles

function getCycles<N, E>(
  graph: Graph<N, E>
): GraphPath<N, E>[]
Returns all elementary cycles as an array of paths. Delegates to genCycles() internally. Complexity: Exponential (can be many cycles in a graph) When to use:
  • Finding all feedback loops
  • Analyzing circular dependencies
  • Graph verification
  • Identifying problematic cycles
Note: Elementary cycles are simple cycles where no node is repeated except the start/end node.

​
Example

import { createGraph, getCycles } from '@statelyai/graph';

const graph = createGraph({
  nodes: [{ id: 'a' }, { id: 'b' }],
  edges: [
    { id: 'ab', sourceId: 'a', targetId: 'b' },
    { id: 'ba', sourceId: 'b', targetId: 'a' },
  ],
});

const cycles = getCycles(graph);
// One cycle: a -> b -> a
cycles.forEach(cycle => {
  const path = [cycle.source.id, ...cycle.steps.map(s => s.node.id)];
  console.log('Cycle:', path.join(' -> '));
});

​
Finding feedback loops

const system = createGraph({
  nodes: [
    { id: 'sensor', data: { type: 'input' } },
    { id: 'controller', data: { type: 'logic' } },
    { id: 'actuator', data: { type: 'output' } },
  ],
  edges: [
    { id: '1', sourceId: 'sensor', targetId: 'controller' },
    { id: '2', sourceId: 'controller', targetId: 'actuator' },
    { id: '3', sourceId: 'actuator', targetId: 'sensor' }, // feedback
  ],
});

const cycles = getCycles(system);
console.log(`Found ${cycles.length} feedback loop(s)`);

cycles.forEach(cycle => {
  const nodes = [cycle.source, ...cycle.steps.map(s => s.node)];
  console.log('Feedback loop:', nodes.map(n => n.id).join(' -> '));
});

​
Undirected graphs

For undirected graphs, cycles are found by treating each edge as bidirectional: 
const graph = createGraph({
  type: 'undirected',
  nodes: [{ id: 'a' }, { id: 'b' }, { id: 'c' }],
  edges: [
    { id: 'ab', sourceId: 'a', targetId: 'b' },
    { id: 'bc', sourceId: 'b', targetId: 'c' },
    { id: 'ca', sourceId: 'c', targetId: 'a' },
  ],
});

const cycles = getCycles(graph);
// One cycle: a -> b -> c -> a

​
genCycles

function genCycles<N, E>(
  graph: Graph<N, E>
): Generator<GraphPath<N, E>>
Lazily yields elementary cycles one at a time. Use getCycles() for the full array. When to use:
  • Processing cycles without loading all into memory
  • Early termination when finding problematic cycle
  • Large graphs with many cycles

​
Example

import { createGraph, genCycles } from '@statelyai/graph';

const graph = createGraph({
  nodes: [{ id: 'a' }, { id: 'b' }],
  edges: [
    { id: 'ab', sourceId: 'a', targetId: 'b' },
    { id: 'ba', sourceId: 'b', targetId: 'a' },
  ],
});

for (const cycle of genCycles(graph)) {
  console.log('Found cycle:', cycle.steps.map(s => s.node.id));
  // ['b', 'a']
}

​
Finding first cycle

function findFirstCycle(graph) {
  for (const cycle of genCycles(graph)) {
    return cycle; // Return immediately
  }
  return null;
}

const firstCycle = findFirstCycle(graph);
if (firstCycle) {
  console.log('Found a cycle - graph is not acyclic');
}

​
Use Cases

​
Dependency validation

Validate that module imports don’t create circular dependencies: 
import { createGraph, getCycles } from '@statelyai/graph';

function validateDependencies(imports: Map<string, string[]>) {
  const graph = createGraph({
    nodes: Array.from(imports.keys()).map(id => ({ id })),
    edges: Array.from(imports.entries()).flatMap(([source, targets]) =>
      targets.map((target, i) => ({
        id: `${source}-${target}-${i}`,
        sourceId: source,
        targetId: target,
      }))
    ),
  });

  const cycles = getCycles(graph);
  
  if (cycles.length > 0) {
    console.error('Circular dependencies detected:');
    cycles.forEach(cycle => {
      const path = [cycle.source.id, ...cycle.steps.map(s => s.node.id)];
      console.error('  ', path.join(' -> '));
    });
    return false;
  }
  
  return true;
}

​
State machine analysis

Find cycles in state transitions: 
const stateMachine = createGraph({
  nodes: [
    { id: 'idle' },
    { id: 'loading' },
    { id: 'success' },
    { id: 'error' },
  ],
  edges: [
    { id: '1', sourceId: 'idle', targetId: 'loading' },
    { id: '2', sourceId: 'loading', targetId: 'success' },
    { id: '3', sourceId: 'loading', targetId: 'error' },
    { id: '4', sourceId: 'error', targetId: 'idle' }, // retry cycle
    { id: '5', sourceId: 'success', targetId: 'idle' }, // reset cycle
  ],
});

const cycles = getCycles(stateMachine);
console.log(`State machine has ${cycles.length} cycle(s)`);

cycles.forEach(cycle => {
  const states = [cycle.source.id, ...cycle.steps.map(s => s.node.id)];
  console.log('Cycle:', states.join(' -> '));
  // Example: idle -> loading -> error -> idle
});

​
Deadlock detection

Find potential deadlocks in resource allocation: 
const resources = createGraph({
  nodes: [
    { id: 'thread1', data: { type: 'thread' } },
    { id: 'thread2', data: { type: 'thread' } },
    { id: 'lock1', data: { type: 'lock' } },
    { id: 'lock2', data: { type: 'lock' } },
  ],
  edges: [
    // thread1 holds lock1, wants lock2
    { id: '1', sourceId: 'lock1', targetId: 'thread1' },
    { id: '2', sourceId: 'thread1', targetId: 'lock2' },
    // thread2 holds lock2, wants lock1
    { id: '3', sourceId: 'lock2', targetId: 'thread2' },
    { id: '4', sourceId: 'thread2', targetId: 'lock1' },
  ],
});

const cycles = getCycles(resources);
if (cycles.length > 0) {
  console.error('Potential deadlock detected!');
  cycles.forEach(cycle => {
    const entities = [cycle.source.id, ...cycle.steps.map(s => s.node.id)];
    console.error('Deadlock cycle:', entities.join(' -> '));
  });
}

​
Comparison with isAcyclic

isAcyclic() is much faster than getCycles() when you only need to know if cycles exist: 
import { isAcyclic, getCycles } from '@statelyai/graph';

// Fast: O(V + E)
if (!isAcyclic(graph)) {
  console.log('Graph has cycles');
}

// Slow: exponential
const cycles = getCycles(graph);
if (cycles.length > 0) {
  console.log('Graph has cycles');
}
Use getCycles() only when you need to know what the cycles are, not just whether they exist.

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