According to Lemma 22.11 of Cormen et al., Introduction to Algorithms (CLRS):
A directed graph G is acyclic if and only if a depth-first search of G yields no back edges.
This has been mentioned in several answers; here I'll also provide a code example based on chapter 22 of CLRS. The example graph is illustrated below.
CLRS' pseudo-code for depth-first search reads:
In the example in CLRS Figure 22.4, the graph consists of two DFS trees: one consisting of nodes u, v, x, and y, and the other of nodes w and z. Each tree contains one back edge: one from x to v and another from z to z (a self-loop).
The key realization is that a back edge is encountered when, in the DFS-VISIT function, while iterating over the neighbors v of u, a node is encountered with the GRAY color.
The following Python code is an adaptation of CLRS' pseudocode with an if clause added which detects cycles:
import collections
class Graph(object):
def __init__(self, edges):
self.edges = edges
self.adj = Graph._build_adjacency_list(edges)
@staticmethod
def _build_adjacency_list(edges):
adj = collections.defaultdict(list)
for edge in edges:
adj[edge[0]].append(edge[1])
return adj
def dfs(G):
discovered = set()
finished = set()
for u in G.adj:
if u not in discovered and u not in finished:
discovered, finished = dfs_visit(G, u, discovered, finished)
def dfs_visit(G, u, discovered, finished):
discovered.add(u)
for v in G.adj[u]:
# Detect cycles
if v in discovered:
print(f"Cycle detected: found a back edge from {u} to {v}.")
# Recurse into DFS tree
if v not in finished:
dfs_visit(G, v, discovered, finished)
discovered.remove(u)
finished.add(u)
return discovered, finished
if __name__ == "__main__":
G = Graph([
('u', 'v'),
('u', 'x'),
('v', 'y'),
('w', 'y'),
('w', 'z'),
('x', 'v'),
('y', 'x'),
('z', 'z')])
dfs(G)
Note that in this example, the time in CLRS' pseudocode is not captured because we're only interested in detecting cycles. There is also some boilerplate code for building the adjacency list representation of a graph from a list of edges.
When this script is executed, it prints the following output:
Cycle detected: found a back edge from x to v.
Cycle detected: found a back edge from z to z.
These are exactly the back edges in the example in CLRS Figure 22.4.
