Abstract
Two graphs G and H are homomorphism indistinguishable over a class of graphs ℱ if for all graphs F ∈ ℱ the number of homomorphisms from F to G is equal to the number of homomorphisms from F to H. Many natural equivalence relations comparing graphs 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 selfcomplementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minorclosed graph class. Selfcomplementarity is a mild property satisfied by most wellstudied logics. This result follows from a correspondence between closure properties of a graph class and preservation properties of its homomorphism indistinguishability relation.
Furthermore, we classify all graph classes which are in a sense finite (essentially profinite) 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 general properties of the homomorphism distinguishing closure.
BibTeX  Entry
@InProceedings{seppelt:LIPIcs.MFCS.2023.82,
author = {Seppelt, Tim},
title = {{Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors}},
booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
pages = {82:182:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772921},
ISSN = {18688969},
year = {2023},
volume = {272},
editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18616},
URN = {urn:nbn:de:0030drops186161},
doi = {10.4230/LIPIcs.MFCS.2023.82},
annote = {Keywords: homomorphism indistinguishability, graph minor, logic}
}
Keywords: 

homomorphism indistinguishability, graph minor, logic 
Collection: 

48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023) 
Issue Date: 

2023 
Date of publication: 

21.08.2023 