The Descriptive Complexity of Subgraph Isomorphism without Numerics

TL;DR

This paper investigates the descriptive complexity of subgraph isomorphism in first-order logic, revealing bounds on quantifier depth related to graph properties like size, treewidth, and connectivity, with exact results for all 4-vertex graphs.

Contribution

It establishes new bounds on the quantifier depth needed to define subgraph isomorphism, connecting logical definability with graph parameters such as treewidth and density.

Findings

Quantifier depth can be as low as  for some graphs in large connected graphs.

Quantifier depth cannot be better than half the size of the pattern graph.

Exact complexity parameters are determined for all connected graphs on 4 vertices.

Abstract

Let $F$ be a connected graph with $\ell$ vertices. The existence of a subgraph isomorphic to $F$ can be defined in first-order logic with quantifier depth no better than $\ell$, simply because no first-order formula of smaller quantifier depth can distinguish between the complete graphs $K_\ell$ and $K_{\ell-1}$. We show that, for some $F$, the existence of an $F$ subgraph in \emph{sufficiently large} connected graphs is definable with quantifier depth $\ell-3$. On the other hand, this is never possible with quantifier depth better than $\ell/2$. If we, however, consider definitions over connected graphs with sufficiently large treewidth, the quantifier depth can for some $F$ be arbitrarily small comparing to $\ell$ but never smaller than the treewidth of $F$. Moreover, the definitions over highly connected graphs require quantifier depth strictly more than the density of $F$. Finally, we determine the exact values of these descriptive complexity parameters for all connected pattern graphs $F$ on 4 vertices.