Structural measures of graphs, such as treewidth, are central tools in computational complexity resulting in efficient algorithms when exploiting the parameter. It is even known that modern SAT solvers work efficiently on instances of small treewidth. Since these solvers are widely applied in knowledge representation reasoning and symbolic AI, research interests in compact encodings for solving and to understand encoding limitations. Even more general is the graph parameter clique-width, which unlike treewidth can be small for dense graphs, e.g., co-graphs. Although dynamic programming algorithms and logic-based characterizations are available for clique-width, little is known about encodings. In this work, we initiate the quest to understand encoding capabilities with clique-width by considering abstract argumentation, which is a robust knowledge representation framework widely used for reasoning with conflicting arguments. It is based on directed graphs and asks for computationally challenging properties, making it a natural candidate to study computational properties. We design novel reductions from argumentation problems to (Q)SAT. Our reductions linearly preserve the clique-width, resulting in directed decomposition-guided (DDG) reductions. Thereby, we establish novel results for all argumentation semantics, including counting. We show that the overhead caused by our DDG reductions cannot be significantly improved under reasonable assumptions thereby providing a structurally optimal reduction contributing to the encoding theory of argumentation.