Abstract
Let φ(F) be the property of containing (as a subgraph) an isomorphic copy of a graph F. It is easy to show that this property cannot be defined in a first-order language by a sentence with a quantifier depth (or variable width) strictly less than the number of vertices in F. Nevertheless, such a definition exists in some classes of graphs. Three classes of graphs are considered: connected graphs with a large number of vertices, graphs with large treewidth, and graphs with high connectivity.
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