  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
generalise the notion of a delay as defined by Drinen 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.
