Complementary Graph Entropy, AND Product, and Disjoint Union of Graphs
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In the zero-error Slepian-Wolf source coding problem, the optimal rate is given by the complementary graph entropy 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 ∧, and as a disjoint union ⊔. We derive a structural result that equates H(∧ ·) and H(⊔ ·) up to a multiplicative constant, which has two consequences. First, we prove that the cases where H(∧ ·) and H(⊔ ·) can be linearized coincide. Second, we determine H in cases where it was unknown: products of perfect graphs; and G_5 ∧ G when G is a perfect graph, using Tuncel et al.'s result for H(G_5 ⊔ G). The graphs in these cases are not perfect in general.
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