The Descriptive Complexity of Subgraph Isomorphism Without Numerics

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 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.