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Directed acyclic graphs (dags) are often used to model circuits. Path lengths in such dags represent circuit delays. In the vertex splitting problem, the objective is to determine a minimum number of vertices to split so that the resulting dag has no path of length > δ. This problem has application to the placement of flip-flops in partial scan designs, placement of latches in pipelined circuits, placement of signal boosters in lossy circuits and networks, etc. Several simplified versions of this problem are shown to be NP-hard. A linear time algorithm is obtained for the case when the dag is a tree. A backtracking algorithm and heuristics are developed for general dags and experimental results using dags obtained from ISCAS benchmark circuits are obtained.