Subsets of vertices give Morita equivalences of Leavitt path algebras

We show that every subset of vertices of a directed graph E gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an algebraic version of a theorem of Crisp and Gow: certain subgraphs of E can be contracted to a new graph G such that the Leavitt path algebras of E and G are Morita equivalent. We provide examples to illustrate how desingularising a graph, and in- or out-delaying of a graph, all fit into this setting.