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An optimal lower bound on the number of variables for graph identification

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Abstract

In this paper we show that Ω(n) variables are needed for first-order logic with counting to identify graphs onn vertices. Thek-variable language with counting is equivalent to the (k−1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant becausen variables obviously suffice to identify graphs onn vertices.

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Authors and Affiliations

  1. Computer Science Dept., Princeton University, 08540, Princeton, NJ

    Jin-Yi Cai

  2. Computer Science Dept., University of Massachusetts, 01003, Amherst, MA

    Martin Fürer

  3. Computer Science Dept., Pennsylvania State University, 16802, University Park, PA

    Neil Immerman

Authors
  1. Jin-Yi Cai
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  2. Martin Fürer
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  3. Neil Immerman
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Additional information

Research supported by NSF grant CCR-8709818.

Research supported by NSF grant CCR-8805978 and Pennsylvania State University Research Initiation grant 428-45.

Research supported by NSF grants DCR-8603346 and CCR-8806308.

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Cai, JY., Fürer, M. & Immerman, N. An optimal lower bound on the number of variables for graph identification. Combinatorica 12, 389–410 (1992). https://doi.org/10.1007/BF01305232

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