On Algorithms Based on Finitely Many Homomorphism Counts

TL;DR

This paper investigates which graph properties can be decided solely based on finitely many homomorphism counts, characterizing the logical complexity of such properties and establishing the minimal number of counts needed for graph isomorphism.

Contribution

It characterizes the class of graph properties with a hom algorithm, linking them to first-order logic fragments, and determines the minimal number of homomorphism counts needed for graph isomorphism.

Findings

Planarity and 3-colorability lack hom algorithms.

Properties expressible as Boolean combinations of universal FO sentences have hom algorithms.

Three homomorphism counts are necessary and sufficient for graph isomorphism detection.

Abstract

It is well known [Lov\'asz, 67] that up to isomorphism a graph~$G$ is determined by the homomorphism counts $\hom(F, G)$, i.e., the number of homomorphisms from $F$ to $G$, where $F$ ranges over all graphs. Thus, in principle, we can answer any query concerning $G$ with only accessing the $\hom(\cdot,G)$'s instead of $G$ itself. In this paper, we deal with queries for which there is a hom algorithm, i.e., there are finitely many graphs $F_1, \ldots, F_k$ such that for any graph $G$ whether it is a Yes-instance of the query is already determined by the vector\[\overrightarrow{\hom}_{F_1,\ldots,F_k}(G):= \big(\hom(F_1,G),\ldots,\hom(F_k,G)\big),\]where the graphs $F_1, \ldots, F_k$ only depend on $\varphi$. We observe that planarity of graphs and 3-colorability of graphs, properties expressible in monadic second-order logic, have no hom algorithm. On the other hand, queries expressible as a Boolean combination of universal sentences in first-order logic FO have a hom algorithm. Even though it is not easy to find FO definable queries without a hom algorithm, we succeed to show this for the non-existence of an isolated vertex, a property expressible by the FO sentence $\forall x\exists y Exy$, somehow the ``simplest'' graph property not definable by a Boolean combination of universal sentences.These results provide a characterization of the prefix classes of first-order logic with the property that each query definable by a sentence of the prefix class has a hom algorithm. For adaptive query algorithms, i.e., algorithms that again access $\overrightarrow{\hom}_{F_1,\ldots,F_k}(G)$ but here $F_{i+1}$ might depend on $\hom(F_1,G),\ldots,\hom(F_i,G)$, we show that three homomorphism counts $\hom(\cdot,G)$ are both sufficient and in general necessary to determine the isomorphism type of $G$.