## An Optimal Lower Bound on the Number of Variables for Graph Identification (1992)

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Venue: | Combinatorica |

Citations: | 135 - 9 self |

### BibTeX

@ARTICLE{Cai92anoptimal,

author = {Jin-yi Cai and Neil Immerman and Martin Fürer},

title = {An Optimal Lower Bound on the Number of Variables for Graph Identification},

journal = {Combinatorica},

year = {1992},

volume = {12},

pages = {389--410}

}

### Years of Citing Articles

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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 on n vertices. The k-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 because n variables obviously suffice to identify graphs on n vertices. 1

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Citation Context ...ce graphs. Individualization of k = √ n(log n) c vertices splits a bounded valence graph into color classes of size at most √ n resulting in an exp( √ n(log n) c ) isomorphism test. Subsequently Luks =-=[33]-=- proved, using group theory to greater depth, that isomorphism for graphs of bounded valence is in polynomial time. Finally the canonical labeling problem for graphs of bounded valence has been solved... |

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Citation Context ...e presented. 4 Pebbling Games We next describe two pebbling games that are equivalent to testing Lk and Ck equivalence, respectively. These games are variants of the games of Ehrenfeucht and Fraïssé, =-=[16, 17]-=-. The results in this section concerning the Lk game and the Ck game originally appeared in [23] and [27], respectively. Let G and H be two graphs, and let m and k be natural numbers. Define the m-mov... |

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Citation Context ...sm problems of primitive and arbitrary coherent configurations are polynomial time equivalent [11]. The general graph isomorphism problem has been attacked by Zemlyachenko. The method is described in =-=[5]-=- and [41]. By individualization of O( √ n) vertices and canonical edgeswitching, he has been able to reduce the valence to O( √ n). Combining this with the method of Luks [33], Zemlyachenko obtained t... |

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Citation Context ...o Weisfeiler’s book [39]. The Russians have built a huge algebraic theory with extensive applications around the notion of stable colorings of pairs. The key notion is that of a cellular algebra (see =-=[39, 30]-=-), which has been discovered in another context and called coherent configuration by Higman [21]. Weisfeiler and Lehman have asked whether the special k-dim W-L method with a slowly growing value of k... |

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Citation Context ...imensional refinement (i.e., the special W-L method) has produced pioneer results in the areas of bounded valence as well as general graph isomorphism and canonical labeling. Babai’s technical report =-=[3]-=- started to use group theoretical algorithms to obtain provable upper bounds for isomorphism problems. Not only did he get his well known probabilistic polynomial time isomorphism test for graphs of b... |

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Citation Context ...rant 428-45. ‡ Research supported by NSF grants DCR-8603346 and CCR-8806308. 1sAnother way to interpret our results is with stable colorings of k-tuples of vertices. The work of Weisfeiler and Lehman =-=[40, 39]-=- on combinatorial and group theoretic properties of colored graphs, has inspired the idea of separating the orbits of the automorphism group of a graph by coloring k-tuples of vertices. Sometimes, thi... |

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Citation Context ...s around the notion of stable colorings of pairs. The key notion is that of a cellular algebra (see [39, 30]), which has been discovered in another context and called coherent configuration by Higman =-=[21]-=-. Weisfeiler and Lehman have asked whether the special k-dim W-L method with a slowly growing value of k would be sufficient to solve the graph isomorphism problem. There was actually good reason to c... |

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Citation Context ...ounded valence has been solved in polynomial time by [11] and [18] independently. Individualization followed by naive refinement has also been the tool used by Babai to handle strongly regular graphs =-=[4]-=- and primitive coherent configurations [6]. He used individualization of k = 2 √ n log n vertices. Strongly regular graphs and more generally, 4scoherent configurations are stable under 2-dim W-L. Whi... |

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Citation Context ...o the color of (u, v). The algorithm stops when no color class is split any more. A modification of this algorithm has been shown to produce normal forms for all regular graphs in linear average time =-=[29]-=-. The k-tuple coloring algorithm (named k-dim W-L by Babai [13]) classifies k-tuples of vertices. It might color vertices and edges implicitly by using k-tuples with repetition of components. It could... |

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3 |
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Citation Context ...reater depth, that isomorphism for graphs of bounded valence is in polynomial time. Finally the canonical labeling problem for graphs of bounded valence has been solved in polynomial time by [11] and =-=[18]-=- independently. Individualization followed by naive refinement has also been the tool used by Babai to handle strongly regular graphs [4] and primitive coherent configurations [6]. He used individuali... |

1 | The Expressiveness of a Family of Finite - Immerman, Patnaik, et al. - 1991 |