The first order definability of graphs: upper bounds for quantifier depth

  • Authors:

  • Oleg Pikhurko;Helmut Veith;Oleg Verbitsky

  • Affiliations:

  • Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA;Institut für Informatik, TU München, Germany;Department of Mechanics and Mathematics, Kyiv University, Ukraine

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2006

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Abstract

Let D(G) denote the minimum quantifier depth of a first order sentence that defines a graph G up to isomorphism in terms of the adjacency and equality relations. Call two vertices of G similar if they have the same adjacency to any other vertex and denote the maximum number of pairwise similar vertices in G by σ(G). We prove that σ(G) + 1 ≤ D(G) ≤ max[σ(G) + 2, {n + 5)/2}, where n denotes the number of vertices of G. In particular, D(G) ≤ (n + 5)/2 for every G with no transposition in the automorphism group. If G is connected and has maximum degree d, we prove that D(G) ≤ cdn + O(d2) for a constant cd D(G) hold true even if We allow only definitions with at most one alternation in any sequence of nested quantifiers.In passing we establish an upper bound for a related number D(G, G'), the minimum quantifier depth of a first order sentence which is true on exactly one of graphs G and G'. If G and G' are non-isomorphic and both have n vertices, then D (G, G') ≤ (n + 3)/2. This bound is tight up to an additive constant of I. If we additionally require that a sentence distinguishing G and G' is existential, we prove only a slightly weaker bound D(G, G') ≤ (n + 5)/2.