Complementary Graph Entropy, AND Product, and Disjoint Union of Graphs

In the zero-error Slepian-Wolf source coding problem, the optimal rate is given by the complementary graph entropy $\bar H$ of the characteristic graph. It has no single-letter formula, except for perfect graphs, for the pentagon graph with uniform distribution G5, and for their disjoint union. We consider two particular instances, where the characteristic graphs respectively write as an AND product ∧, and as a disjoint union ⊔. We derive a structural result that equates $\bar H( \wedge )$ and $\bar H( \sqcup )$ up to a multiplicative constant, which has two consequences. First, we prove that the cases where $\bar H( \wedge )$ and $\bar H( \sqcup )$ can be linearized coincide. Second, we determine $\bar H$ in cases where it was unknown: products of perfect graphs; and G5 ∧ G when G is a perfect graph, using Tuncel et al.’s result for $\bar H({G_5} \sqcup G)$. The graphs in these cases are not perfect in general.