This paper explores the effect of various graphical constructions upon the associated graph C∗-algebras. The graphical constructions in question arise naturally in the study of flow equivalence for topological Markov chains. We prove that out-splittings give rise to isomorphic graph algebras, and in-splittings give rise to strongly Morita equivalent C∗-algebras. We generalize the notion of a delay as defined in (D. Drinen, Preprint, Dartmouth College, 2001) to form in-delays and out-delays. We prove that these constructions give rise to Morita equivalent graph C∗-algebras. We provide examples which suggest that our results are the most general possible in the setting of the C∗-algebras of arbitrary directed graphs.