The First Order Definability of Graphs: Upper Bounds for Quantifier Rank

We say that a first order formula Φ distinguishes a graph G from another graph G′ if Φ is true on G and false on G′. Provided G and G′ are non-isomorphic, let D(G,G′) denote the minimal quantifier rank of a such formula. Let n denote the order of G. We prove that, if G′ has the same order, then D(G,G′) ≤ (n+3)/2. This bound is tight up to an additive constant of 1. Furthermore, we prove that non-isomorphic G and G′ of order n are distinguishable by an existential formula of quantifier rank at most (n+5)/2. As a consequence of the first result, we obtain an upper bound of (n+1)/2 for the optimum dimension of the Weisfeiler-Lehman graph canonization algorithm, whose worst case value is known to be linear in n. We say that a first order formula Φ defines a graph G if Φ distinguishes G from every non-isomorphic graph G′. Let D(G) be the minimal quantifier rank of a formula defining G. As it is well known, D(G) ≤ n + 1 and this bound is generally best possible. Nevertheless, we here show that there is a class C of graphs of simple, easily recognizable structure such that • D(G) ≤ (n + 5)/2 with the exception of all graphs in C; • if G ∈ C, then it is easy to compute the exact value of D(G). Moreover, the defining formulas in this result have only one quantifier alternation. The bound for D(G) can be improved for graphs with bounded vertex degrees: For each d ≥ 2 there is a constant cd < 1/2 such that D(G) ≤ cdn + O(d2) for any graph G with no isolated vertices and edges whose maximum degree is d. Finally, we extend our results over directed graphs, more generally, over arbitrary structures with maximum relation arity 2, and over k-uniform hypergraphs. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890 Web: http://www.math.cmu.edu/~pikhurko/ Institut für Informationssysteme, Technische Universität Wien, Favoritenstr. 9, A-1040 Wien, Austria. Supported by the European Community Research Training Network “Games and Automata for Synthesis and Validation” (GAMES) and by the Austrian Science Fund Project Z29INF. E-mail: [email protected] Department of Mechanics &Mathematics, Kyiv University, Ukraine. Research was done in part while visiting the Institut für Informationssysteme at the Technische Universität Wien, supported from Austrian Science Foundation grant Z29-INF. E-mail: [email protected]