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# Homomorphism-Distinguishing Closedness for Graphs of Bounded Tree-Width

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LIPIcs.STACS.2024.53.pdf
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## Acknowledgements

I want to thank Tim Seppelt for pointing me to Lemma 4 in [Seppelt, MFCS 2023].

## Cite As

Daniel Neuen. Homomorphism-Distinguishing Closedness for Graphs of Bounded Tree-Width. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 53:1-53:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.STACS.2024.53

## Abstract

Two graphs are homomorphism indistinguishable over a graph class 𝐅, denoted by G ≡_𝐅 H, if hom(F,G) = hom(F,H) for all F ∈ 𝐅 where hom(F,G) denotes the number of homomorphisms from F to G. A classical result of Lovász 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 𝐅 is homomorphism-distinguishing closed if, for every F ∉ 𝐅, there are graphs G and H such that G ≡_𝐅 H and hom(F,G) ≠ 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 𝒯_k, k ≥ 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].

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph theory
• Mathematics of computing → Combinatorial algorithms
• Theory of computation → Graph algorithms analysis
##### Keywords
• homomorphism indistinguishability
• tree-width
• Weisfeiler-Leman algorithm
• subgraph counts

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