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.

中文翻译：

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有向图可还原性

我们引入了有向图可归约的概念，它是图可归约的相对版本。有向图可约性具有很强的群论和拓扑结果。给出了存在有向图可还原性的 Freiheitssatz 的多关系版本。显示了关于子配合物的非球面性和 $\pi_1$-injectivity 的结果。我们将图解可还原性的 Corson-Trace 表征概括为有向图可还原性。我们将相对表示的图解可还原性与定向图解可还原性进行比较。用于显示图解可简化性的经典工具，例如权重测试、最大/最小测试和小消除技术，适用于定向图可简化。