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The Descriptive Complexity of Subgraph Isomorphism Without Numerics

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Computer Science – Theory and Applications (CSR 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10304))

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Abstract

Let F be a connected graph with \(\ell \) vertices. The existence of a subgraph isomorphic to F can be defined in first-order logic with quantifier depth no better than \(\ell \), simply because no first-order formula of smaller quantifier depth can distinguish between the complete graphs \(K_\ell \) and \(K_{\ell -1}\). We show that, for some F, the existence of an F subgraph in sufficiently large connected graphs is definable with quantifier depth \(\ell -3\). On the other hand, this is never possible with quantifier depth better than \(\ell /2\). If we, however, consider definitions over connected graphs with sufficiently large treewidth, the quantifier depth can for some F be arbitrarily small comparing to \(\ell \) but never smaller than the treewidth of F.

We also prove that any first-order definition of the existence of an induced subgraph isomorphic to F requires quantifier depth strictly more than the density of F, even over highly connected graphs. From this bound we derive a succinctness result for existential monadic second-order logic: A usage of just one monadic quantifier sometimes reduces the first-order quantifier depth at a super-recursive rate.

The first author was supported by DFG grant VE 652/1–2. He is on leave from the IAPMM, Lviv. The second author was supported by grants No. 15-01-03530 and 16-31-60052 of Russian Foundation for Basic Research.

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Notes

  1. 1.

    In his presentation [20], Benjamin Rossman states upper bounds \(W_\mathrm {FO} (F)\le tw (F)+1\) and \(D_\mathrm {FO} (F)\le td (F)\) for the colorful version of Subgraph Isomorphism studied in [13]. It is not hard to observe that the auxiliary color predicates can be defined in \(\mathrm {FO} [\mathrm {Arb}]\) at the cost of two extra quantified variables by the color-coding method developed in [2]; see also [3, Theorem 4.2].

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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 also thank the anonymous referee who provided us with numerous useful comments on the manuscript.

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Authors and Affiliations

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

    Oleg Verbitsky

  2. Department of Discrete Mathematics, Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow Region, Russia

    Maksim Zhukovskii

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  1. Oleg Verbitsky
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  2. Maksim Zhukovskii
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Correspondence to Oleg Verbitsky .

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  1. University of Bordeaux , Talence, France

    Pascal Weil

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Verbitsky, O., Zhukovskii, M. (2017). The Descriptive Complexity of Subgraph Isomorphism Without Numerics. In: Weil, P. (eds) Computer Science – Theory and Applications. CSR 2017. Lecture Notes in Computer Science(), vol 10304. Springer, Cham. https://doi.org/10.1007/978-3-319-58747-9_27

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