Tight Bounds on the Asymptotic Descriptive Complexity of Subgraph Isomorphism

Abstract

Let v ( F ) denote the number of vertices in a fixed connected pattern graph F . We show an infinite family of patterns F such that the existence of a subgraph isomorphic to F is expressible by a first-order sentence of quantifier depth 2/3 v ( F ) + 1, assuming that the host graph is sufficiently large and connected. However, this is impossible for any F using less than 2/3 v ( F ) - 2 first-order variables.