  We introduce the notion of directed diagrammatic reducibility which is a
relative version of diagrammatic reducibility. Directed diagrammatic
reducibility has strong group theoretic and topological consequences. A
multi-relator version of the Freiheitssatz in the presence of directed
diagrammatic reducibility is given. Results concerning asphericity and
$\pi_1$-injectivity of subcomplexes are shown. We generalize the Corson-Trace
characterization of diagrammatic reducibility to directed diagrammatic
reducibility. We compare diagrammatic reducibility of relative presentations to
directed diagrammatic reducibility. Classical tools for showing diagrammatic
reducibility, such as the weight test, the max/min test, and small cancellation
techniques are adapted to directed diagrammatic reducibility. The paper ends
with some applications to labeled oriented trees.
