The First Order Definability of Graphs: Upper Bounds for Quantifier Rank
Combinatorics
2016-09-07 v1 Logic
Abstract
We say that a first order formula A distinguishes a graph G from another graph G' if A 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. We prove that, if G and G' have the same order n, then D(G,G')\le(n+3)/2, which is tight up to an additive constant of 1. The analogous questions are considered for directed graphs (more generally, for arbitrary structures with maximum relation arity 2) and for k-uniform hypergraphs. Also, we study defining formulas, where we require that A distinguishes G from any other non-isomorphic G'.
Cite
@article{arxiv.math/0311041,
title = {The First Order Definability of Graphs: Upper Bounds for Quantifier Rank},
author = {Oleg Pikhurko and Helmut Veith and Oleg Verbitsky},
journal= {arXiv preprint arXiv:math/0311041},
year = {2016}
}
Comments
52 pages