The directed \(\overrightarrow{P}_k\)-graph of a digraph \(D\) is obtained by representing the directed paths on \(k\) vertices of \(D\) by vertices. Two such vertices are joined by an arc whenever the corresponding directed paths in \(D\) form a directed path on \(k+1\) vertices or a directed cycle on \(k\) vertices in \(D\). In this paper, we give a necessary and sufficient condition for two digraphs with isomorphic \(\overrightarrow{P}_3\)-graphs. This improves a previous result, where some additional conditions were imposed.
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