The Descriptive Complexity of Subgraph Isomorphism Without Numerics

Let F be a connected graph with ℓ vertices. The existence of a subgraph isomorphic to F can be defined in first-order logic with quantifier depth no better than ℓ, simply because no first-order formula of smaller quantifier depth can distinguish between the complete graphs Kℓ and Kℓ− 1. We show that, for some F, the existence of an F subgraph in sufficiently large connected graphs is definable with quantifier depth ℓ − 3. On the other hand, this is never possible with quantifier depth better than ℓ/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 ℓ 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.