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The Descriptive Complexity of Subgraph Isomorphism Without Numerics

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Abstract

Let F be a connected graph with ℓ vertices. The existence of a subgraph isomorphic to F can be defined in first-order logic with quantifier depth no better than ℓ, simply because no first-order formula of smaller quantifier depth can distinguish between the complete graphs Kℓ and Kℓ− 1. We show that, for some F, the existence of an F subgraph in sufficiently large connected graphs is definable with quantifier depth ℓ − 3. On the other hand, this is never possible with quantifier depth better than ℓ/2. If we, however, consider definitions over connected graphs with sufficiently large treewidth, the quantifier depth can for some F be arbitrarily small comparing to ℓ but never smaller than the treewidth of F. Moreover, the definitions over highly connected graphs require quantifier depth strictly more than the density of F. Finally, we determine the exact values of these descriptive complexity parameters for all connected pattern graphs F on 4 vertices.

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Notes

  1. In his presentation [29], Benjamin Rossman states upper bounds WFO(F) ≤tw(F) + 1 and DFO(F) ≤ td(F) for the colorful version of Subgraph Isomorphism studied in [21]. It is not hard to observe that the auxiliary color predicates can be defined in FO[Arb] at the cost of two extra quantified variables by the color-coding method developed in [2]; see also [3, Thm. 4.2].

  2. Note that our language does not contain the truth constants ⊤ and ⊥; otherwise we would have Dv(P3) = 0.

  3. Very recently [34], we were able to settle this problem by showing that \(D_{v}(F)\le \frac 23\,v(F)+ 1\) for infinitely many pattern graphs F and \(W_{v}(F)>\frac 23\,v(F)-2\) for all F.

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Acknowledgements

We would like to thank Tobias Müller for his kind hospitality during the Workshop on Logic and Random Graphs in the Lorentz Center (August 31 – September 4, 2015), where this work was originated. We are also thankful to the anonymous referees for a number of useful suggestions, in particular, for a simplification in the proof of Lemma 6.2.

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Authors and Affiliations

  1. Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099, Berlin, Germany

    Oleg Verbitsky

  2. Laboratory of Advanced Combinatorics and Network Applications, Moscow Institute of Physics and Technology, Moscow, Russia

    Maksim Zhukovskii

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  1. Oleg Verbitsky
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Correspondence to Oleg Verbitsky.

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This article is part of the Topical Collection on Computer Science Symposium in Russia

O. Verbitsky is supported by DFG grants VE 652/1–2 and KO 1053/8–1. M. Zhukovskii is supported by grants No. 15-01-03530 and 16-31-60052 of Russian Foundation for Basic Research.

Oleg Verbitsky is on leave from the IAPMM, Lviv, Ukraine.

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Verbitsky, O., Zhukovskii, M. The Descriptive Complexity of Subgraph Isomorphism Without Numerics. Theory Comput Syst 63, 902–921 (2019). https://doi.org/10.1007/s00224-018-9864-3

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