Abstract
It is well known [L. Lovász, 1967] that up to isomorphism a graph G is determined by the homomorphism counts hom(F, G), i.e., by the number of homomorphisms from F to G where F ranges over all graphs. Moreover, it suffices that F ranges over the graphs with at most as many vertices as G. Thus, in principle, we can answer any query concerning G with only accessing the hom(⋅, 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 F₁, …, F_k such that for any graph G whether it is a Yesinstance of the query is already determined by the vector hom^⟶_{F₁, …, F_k}(G): = (hom(F₁, G), …, hom(F_k, G)).
We observe that planarity of graphs and 3colorability of graphs, properties expressible in monadic secondorder logic, have no hom algorithm. On the other hand, queries expressible as a Boolean combination of universal sentences in firstorder 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 nonexistence of an isolated vertex, a property expressible by the FO sentence ∀ x∃ 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 firstorder logic with the property that each query definable by a sentence of the prefix class has a hom algorithm.
For adaptive hom algorithms, i.e., algorithms that might access a hom(F_{i+1}, G) with F_{i+1} depending on hom(F_j, G) for 1 ≤ j ≤ i we show that three homomorphism counts hom(⋅, G) are sufficient and in general necessary to determine the (isomorphism type of) G. In particular, by three adaptive queries we can answer any question on G. Moreover, adaptively accessing two hom(⋅, G)’s is already enough to detect an isolated vertex.
In 1993 Chaudhuri and Vardi [S. Chaudhuri and M. Y. Vardi, 1993] showed the analogue of the Lovász Isomorphism Theorem for the right homomorphism vector of a graph G, i.e, the vector of values hom(G,F) where F ranges over all graphs characterizes the isomorphism type of G. We study to what extent our results carry over to the right homomorphism vector.
BibTeX  Entry
@InProceedings{chen_et_al:LIPIcs.MFCS.2022.32,
author = {Chen, Yijia and Flum, J\"{o}rg and Liu, Mingjun and Xun, Zhiyang},
title = {{On Algorithms Based on Finitely Many Homomorphism Counts}},
booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages = {32:132:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772563},
ISSN = {18688969},
year = {2022},
volume = {241},
editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16830},
URN = {urn:nbn:de:0030drops168301},
doi = {10.4230/LIPIcs.MFCS.2022.32},
annote = {Keywords: homomorphism numbers, hom algorithms, adaptive hom algorithms}
}
Keywords: 

homomorphism numbers, hom algorithms, adaptive hom algorithms 
Collection: 

47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022) 
Issue Date: 

2022 
Date of publication: 

22.08.2022 