Logical equivalences, homomorphism indistinguishability, and forbidden minors

Two graphs G and H are homomorphism indistinguishable over a graph class $F$ if for all graphs $F\in F$ the number of homomorphisms from F to G is equal to the number of homomorphisms from F to H. Many graph isomorphism relaxations such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes.

Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics.

Furthermore, we classify all graph classes which are in a sense finite and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various questions raised by Roberson (2022) on properties of the homomorphism distinguishing closure.