In the zero-error Slepian-Wolf source coding problem, the optimal rate is given by the complementary graph entropy $\overline{H}$ of the characteristic graph. It has no single-letter formula, except for perfect graphs, for the pentagon graph with uniform distribution $G_5$, and for their disjoint union. We consider two particular instances, where the characteristic graphs respectively write as an AND product $\wedge$, and as a disjoint union $\sqcup$. We derive a structural result that equates $\overline{H}(\wedge \: \cdot)$ and $\overline{H}(\sqcup \: \cdot)$ up to a multiplicative constant, which has two consequences. First, we prove that the cases where $\overline{H}(\wedge \:\cdot)$ and $\overline{H}(\sqcup \: \cdot)$ can be linearized coincide. Second, we determine $\overline{H}$ in cases where it was unknown: products of perfect graphs; and $G_5 \wedge G$ when $G$ is a perfect graph, using Tuncel et al.'s result for $\overline{H}(G_5 \sqcup G)$. The graphs in these cases are not perfect in general.

翻译：在零误差Slepian-Wolf信源编码问题中，最优码率由特征图的互补图熵$\overline{H}$给出。除完美图、均匀分布五边形图$G_5$及其互斥并外，该量不存在单字母公式。本文考虑两类特例：特征图分别表示为AND乘积$\wedge$与互斥并$\sqcup$。我们推导出一个结构性质，将$\overline{H}(\wedge \:\cdot)$与$\overline{H}(\sqcup \:\cdot)$关联至一个乘法常数，该结果带来两个推论。首先，我们证明$\overline{H}(\wedge \:\cdot)$和$\overline{H}(\sqcup \:\cdot) $可线性化的情形完全一致。其次，利用Tuncel等人关于$\overline{H}(G_5 \sqcup G)$的结论，我们确定了两类未知情形下的$\overline{H}$值：完美图的乘积；以及当$G$为完美图时$G_5 \wedge G$的情形。这些情形中的图通常并非完美图。