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Let $G=(V,E)$ be a directed graph, i.e. $V$ is a finite set and $E\subseteq V\times V$.

We call a subset $J\subseteq V$ extending if for every $v\in V\setminus J$ there is a directed path from some vertex $j\in J$ to $v$.

Is there a polynomial-time algorithm to find an extending subset of minimal size?

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There is a linear time algorithm.

Contract all strongly connected components, then you get a DAG. Find all the source vertex in the DAG. For each source vertex in the DAG, pick a single vertex from the strongly connected component represented by the source vertex.

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