Abstract
We investigate the problem of the most efficient first-order definition of the property of containing an induced subgraph isomorphic to a given pattern graph, which is closely related to the time complexity of the decision problem for this property. We derive a series of new bounds for the minimum quantifier depth of a formula defining this property for pattern graphs on five vertices, as well as for disjoint unions of isomorphic complete multipartite graphs. Moreover, we prove that for any there exists a graph on vertices and a first-order formula of quantifier depth at most that defines the property of containing an induced subgraph isomorphic to this graph. Bibliography: 12 titles.
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