Color Refinement, Homomorphisms, and Hypergraphs

Abstract

Recent results show that the structural similarity of graphs can be characterized by counting homomorphisms to them: the Tree Theorem states that the well-known color-refinement algorithm does not distinguish two graphs G and H if and only if, for every tree T, the number of homomorphisms $\mathsf {Hom}(T, G)$ from T to G is equal to the corresponding number $\mathsf {Hom}(T, H)$ from T to H (Dell, Grohe, Rattan 2018). We show how this approach transfers to hypergraphs by introducing a generalization of color refinement. We prove that it does not distinguish two hypergraphs G and H if and only if, for every connected Berge-acyclic hypergraph B, we have $\mathsf {Hom}(B, G) = \mathsf {Hom}(B, H)$. To this end, we show how homomorphisms of hypergraphs and of a colored variant of their incidence graphs are related to each other. This reduces the above statement to one about vertex-colored graphs.