Tight Bounds on the Asymptotic Descriptive Complexity of Subgraph Isomorphism

TL;DR

This paper establishes tight bounds on the minimal quantifier depth and variable count needed in first-order logic to express subgraph isomorphism for certain pattern graphs, revealing fundamental limits in descriptive complexity.

Contribution

It provides the first precise asymptotic bounds on the descriptive complexity of subgraph isomorphism for an infinite family of patterns.

Findings

Existence of patterns with quantifier depth approximately two-thirds of vertices plus one

Lower bounds show less than two-thirds of vertices minus two variables is impossible

Results reveal fundamental limits in logical expressibility 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 $\frac23\,v(F)+1$, assuming that the host graph is sufficiently large and connected. On the other hand, this is impossible for any $F$ with using less than $\frac23\,v(F)-2$ first-order variables.