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# Fine-Grained Meta-Theorems for Vertex Integrity

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LIPIcs.ISAAC.2021.34.pdf
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## Cite As

Michael Lampis and Valia Mitsou. Fine-Grained Meta-Theorems for Vertex Integrity. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ISAAC.2021.34

## Abstract

Vertex Integrity is a graph measure which sits squarely between two more well-studied notions, namely vertex cover and tree-depth, and that has recently gained attention as a structural graph parameter. In this paper we investigate the algorithmic trade-offs involved with this parameter from the point of view of algorithmic meta-theorems for First-Order (FO) and Monadic Second Order (MSO) logic. Our positive results are the following: (i) given a graph G of vertex integrity k and an FO formula ϕ with q quantifiers, deciding if G satisfies ϕ can be done in time 2^O(k²q + q log q) + n^O(1); (ii) for MSO formulas with q quantifiers, the same can be done in time 2^{2^O(k²+kq)} + n^O(1). Both results are obtained using kernelization arguments, which pre-process the input to sizes 2^O(k²)q and 2^O(k²+kq) respectively. The complexities of our meta-theorems are significantly better than the corresponding meta-theorems for tree-depth, which involve towers of exponentials. However, they are worse than the roughly 2^{O(kq)} and 2^{2^{O(k+q)}} complexities known for corresponding meta-theorems for vertex cover. To explain this deterioration we present two formula constructions which lead to fine-grained complexity lower bounds and establish that the dependence of our meta-theorems on k is best possible. More precisely, we show that it is not possible to decide FO formulas with q quantifiers in time 2^o(k²q), and that there exists a constant-size MSO formula which cannot be decided in time 2^{2^o(k²)}, both under the ETH. Hence, the quadratic blow-up in the dependence on k is unavoidable and vertex integrity has a complexity for FO and MSO logic which is truly intermediate between vertex cover and tree-depth.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Parameterized complexity and exact algorithms
##### Keywords
• Model-Checking
• Fine-grained complexity
• Vertex Integrity

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