[Math] Finding all paths on undirected graph

Question

I have an undirected, unweighted graph, and I'm trying to come up with an algorithm that, given 2 unique nodes on the graph, will find all paths connecting the two nodes, not including cycles. Here's an illustration of what I'd like to do: Graph example

Does this algorithm have a name? Can it be done in polynomial time?

Thanks,

Jesse

Answer 1

Suresh suggested DFS, MRA pointed out that it's not clear that works. Here's my attempt at a solution following that thread of comments. If the graph has $m$ edges, $n$ nodes, and $p$ paths from the source $s$ to the target $t$, then the algorithm below prints all paths in time $O((np+1)(m+n))$. (In particular, it takes $O(m+n)$ time to notice that there is no path.)

The idea is very simple: Do an exhaustive search, but bail early if you've gotten yourself into a corner.

Without bailing early, MRA's counter-example shows that exhaustive search spends $\Omega(n!)$ time even if $p=1$: The node $t$ has only one adjacent edge and its neighbor is node $s$, which is part of a complete (sub)graph $K_{n-1}$.

Push s on the path stack and call search(s):

path // is a stack (initially empty)
seen // is a set

def stuck(x)
   if x == t
     return False
   for each neighbor y of x
     if y not in seen
       insert y in seen
       if !stuck(y)
         return False
   return True

def search(x)
  if x == t
    print path
    return
  seen = set(path)
  if stuck(x)
    return
  for each neighbor y of x
    if y not in path:
      push y on the path
      search(y)
      pop y from the path

Here search does the exhaustive search and stuck could be implemented in DFS style (as here) or in BFS style.