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Flexible queries facilitate, in a novel way, easy and concise querying of databases that have varying structures. Two different semantics, flexible and semiflexible, are introduced and investigated. The complexity of evaluating queries under the two semantics is analyzed. Query evaluation is polynomial in the size of the query, the database and the result in the following two cases. First, a semiflexible DAG query and a tree database. Second, a flexible tree query and a database that is any graph. Query containment and equivalence are also investigated. For the flexible semantics, query equivalence is always polynomial. For the semiflexible semantics, query equivalence is polynomial for DAG queries and exponential when the queries have cycles. Under the semiflexible and flexible semantics, two databases could be equivalent even when they are not isomorphic. Database equivalence is formally defined and …