Graphs Identified by Logics with Counting

We classify graphs and, more generally, finite relational structures that are identified by \({C^{2}}\), that is, two-variable first-order logic with counting. Using this classification, we show that it can be decided in almost linear time whether a structure is identified by \({C^{2}}\). Our classification implies that for every graph identified by this logic, all vertex-colored versions of it are also identified. A similar statement is true for finite relational structures.

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