Decomposing perfect discrete Morse functions on connected sum of 3-manifolds

In this paper, we show that if a closed, connected, oriented 3-manifold M = M 1 # M 2 admits a perfect discrete Morse function, then one can decompose this function as perfect discrete Morse functions on M 1 and M 2 . We also give an explicit construction of a separating sphere on M corresponding to such a decomposition.