# Unique path in a directed graph

I'm designing an algorithm for a class that will determine if a directed graph is unique with respect to a vertex $v$ such that for any $u \ne v$ there is at most one path from $v$ to $u$. I've started by using BFS (breadth-first search) to find the shortest path from v to another vertex u, and then running BFS again to see if an alternate path can be found from v to u. I think this is too time consuming however. Does anyone have any hints as to how the solution can be found with a shorter execution time?

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## migrated from stackoverflow.comSep 24 '12 at 13:49

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## 3 Answers

Use BFS to work backwards from v, flagging each vertex as 'visited' as you go. If you ever hit a vertex you've previously visited, your graph doesn't have the uniqueness property. Otherwise, it does.

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Look at the Max Flow Min Cut algorithm.

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It's a simple modification of any graph traversal: if you find an edge to a previously marked node, then you have multiple paths.

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