Suppose I have a directed graph $G = (V,E)$. Suppose that $v_1$ and $v_2$ are two nodes in the graph. Am I correct the number of simple paths (that is, it has no cycles) from $v_1$ to $v_2$ is $O(E)$? Is it true for the special case of directed acyclic graphs?



That's not true even for DAGs: consider the following, with all edges directed left-to-right:

  o   o   o ... o
 / \ / \ /       \
x   o   o   ...   y
 \ / \ / \       /
  o   o   o ... o

There are $2^{|E|/4}$ paths from $x$ to $y$. In a graph with cycles, it can be even worse: consider a clique.

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  • $\begingroup$ I thank you for your correct response. I think the real problem is that I asked the wrong question. $\endgroup$ – Bob Mar 3 '18 at 16:58
  • $\begingroup$ Ah, well that's a shame. But if you make a new post with the right question, I'll answer that one too, if I can. :-) $\endgroup$ – David Richerby Mar 3 '18 at 20:52

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