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

TL;DR

This paper establishes tight upper bounds on the minimal quantifier rank needed to distinguish non-isomorphic graphs, directed graphs, and hypergraphs using first-order formulas, advancing understanding of their definability complexity.

Contribution

It proves a tight upper bound of (n+3)/2 for the quantifier rank needed to distinguish graphs of order n, extending to directed graphs and hypergraphs, and explores defining formulas.

Findings

D(G,G')eil(n+3)/2 for graphs of order n

Bounds extend to directed graphs and hypergraphs

Provides insights into the complexity of first-order definability

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'.