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# On Homomorphism Indistinguishability and Hypertree Depth

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LIPIcs.ICALP.2024.152.pdf
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## Acknowledgements

We thank Nicole Schweikardt for helpful discussions.

## Cite As

Benjamin Scheidt. On Homomorphism Indistinguishability and Hypertree Depth. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 152:1-152:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.152

## Abstract

GC^k is a logic introduced by Scheidt and Schweikardt (2023) to express properties of hypergraphs. It is similar to first-order logic with counting quantifiers (C) adapted to the hypergraph setting. It has distinct sets of variables for vertices and for hyperedges and requires vertex variables to be guarded by hyperedge variables on every quantification. We prove that two hypergraphs G, H satisfy the same sentences in the logic GC^k with guard depth at most k if, and only if, they are homomorphism indistinguishable over the class of hypergraphs of strict hypertree depth at most k. This lifts the analogous result for tree depth ≤ k and sentences of first-order logic with counting quantifiers of quantifier rank at most k due to Grohe (2020) from graphs to hypergraphs. The guard depth of a formula is the quantifier rank with respect to hyperedge variables, and strict hypertree depth is a restriction of hypertree depth as defined by Adler, Gavenčiak and Klimošová (2012). To justify this restriction, we show that for every H, the strict hypertree depth of H is at most 1 larger than its hypertree depth, and we give additional evidence that strict hypertree depth can be viewed as a reasonable generalisation of tree depth for hypergraphs.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Finite Model Theory
• Mathematics of computing → Hypergraphs
##### Keywords
• homomorphism indistinguishability
• counting logics
• guarded logics
• hypergraphs
• incidence graphs
• tree depth
• elimination forest
• hypertree width

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## References

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