Cycles in an Undirected Graph

Actually, depth first (or indeed breadth first) search isn't quite enough. You need a sightly more complex algorithm.

For instance, suppose there is graph with nodes {a,b,c,d} and edges {(a,b),(b,c),(b,d),(d,c)} where an edge (x,y) is an edge from x to y. (looks something like this, with all edges directed downwards.)

    (a)
     |
     |
    (b)
    / \ 
  (d)  |
   |   |
    \ /
    (c)

Then doing depth first search may visit node (a), then (b), then (c), then backtrack to (b), then visit (d), and finally visit (c) again and conclude there is a cycle -- when there isn't. A similar thing happens with breadth first.

What you need to do is keep track of which nodes your in the middle of visiting. In the example above, when the algorithm reaches (d) it has finished visiting (c) but not (a) or (b). So revisiting a finished node is fine, but visiting an unfinished node means you have a cycle. The usual way to do this is colour each node white(not yet visited), grey(visiting descendants) or black(finished visiting).

here is some pseudo code!

define visit(node n):
  if n.colour == grey: //if we're still visiting this node or its descendants
    throw exception("Cycle found")

  n.colour = grey //to indicate this node is being visited
  for node child in n.children():
    if child.colour == white: //if the child is unexplored
      visit(child)

  n.colour = black //to show we're done visiting this node
  return

then running visit(root_node) will throw an exception if and only if there is a cycle (initially all nodes should be white).