eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-03-11
53:1
53:12
10.4230/LIPIcs.STACS.2024.53
article
Homomorphism-Distinguishing Closedness for Graphs of Bounded Tree-Width
Neuen, Daniel
1
https://orcid.org/0000-0002-4940-0318
University of Bremen, Germany
Two graphs are homomorphism indistinguishable over a graph class π
, denoted by G β‘_π
H, if hom(F,G) = hom(F,H) for all F β π
where hom(F,G) denotes the number of homomorphisms from F to G. A classical result of LovΓ‘sz shows that isomorphism between graphs is equivalent to homomorphism indistinguishability over the class of all graphs. More recently, there has been a series of works giving natural algebraic and/or logical characterizations for homomorphism indistinguishability over certain restricted graph classes.
A class of graphs π
is homomorphism-distinguishing closed if, for every F β π
, there are graphs G and H such that G β‘_π
H and hom(F,G) β hom(F,H). Roberson conjectured that every class closed under taking minors and disjoint unions is homomorphism-distinguishing closed which implies that every such class defines a distinct equivalence relation between graphs. In this work, we confirm this conjecture for the classes π―_k, k β₯ 1, containing all graphs of tree-width at most k.
As an application of this result, we also characterize which subgraph counts are detected by the k-dimensional Weisfeiler-Leman algorithm. This answers an open question from [Arvind et al., J. Comput. Syst. Sci., 2020].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol289-stacs2024/LIPIcs.STACS.2024.53/LIPIcs.STACS.2024.53.pdf
homomorphism indistinguishability
tree-width
Weisfeiler-Leman algorithm
subgraph counts