Compositional graphoids are fundamental discrete structures which appear in probabilistic reasoning, particularly in the area of graphical models. They are semigraphoids which satisfy the Intersection and Composition properties. These important properties, however, are not enjoyed by general probability distributions. This paper surveys what is known about them, providing systematic constructions of examples and counterexamples as well as necessary and sufficient conditions. Novel sufficient conditions for both properties are derived in the context of discrete random variables via information-theoretic tools.

翻译：复合图状结构是概率推理领域（尤其是图模型方向）中基础性的离散结构，它们是满足交集性质与复合性质的半图状结构。然而，一般概率分布并不具备这些重要性质。本文系统综述了相关已知结论，通过信息论工具构建了系统的示例与反例，并给出了充分必要条件。针对离散随机变量，本文推导出了两类性质的新充分条件。