Discrete Applied Mathematics 154 (2006) 2511 – 2529
www.elsevier.com/locate/dam
The first order definability of graphs: Upper bounds for
quantifier depth
Oleg Pikhurko
a
, Helmut Veith
b,1
, Oleg Verbitsky
c,2
a
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA
b
Institut für Informatik, TU München, Germany
c
Department of Mechanics and Mathematics, Kyiv University, Ukraine
Received 14 January 2004; received in revised form 14 November 2005; accepted 14 March 2006
Available online 23 May 2006
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 (G). We prove that (G) + 1 D(G) max{(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) c
d
n + O(d
2
) for a constant c
d
<
1
2
. A linear lower
bound for graphs of maximum degree 3 with no transposition in the automorphism group follows from an earlier result by Cai,
Fürer, and Immerman [An optimal lower bound on the number of variables for graph identification, Combinatorica 12(4) (1992)
389–410]. Our upper bounds for 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
.IfG 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 1. 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.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Graph definability; First order logic; Ehrenfeuct game
1. Introduction
1.1. Statement of the problem and overview of our results
From the logical point of view, a graph G is a structure with a single irreflexive and symmetric binary relation
capturing the vertex adjacency. We consider first order sentences about graphs in the laconic language consisting of
1
Research done in part while affiliated with Institut für Informationssysteme, Technische Universität Wien, 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 Z29-INF.
2
Research was done in part while visiting Institut für Informationssysteme, Technische Universität Wien. Supported by the Austrian Science Fund
Project Z29-INF.
E-mail addresses: veith@in.tum.de (H. Veith), oleg@ov.litech.net (O. Verbitsky).
URL: http://www.math.cmu.edu/∼ pikhurko/ (O. Pikhurko).
0166-218X/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.dam.2006.03.002