In this work the dynamic descriptive complexity of the k-clique query is studied. It is shown that when edges may only be inserted then k-clique can be maintained by a quantifier-free update program of arity k − 1, but it cannot be maintained by a quantifier-free update program of arity k − 2 (even in the presence of unary auxiliary functions). This establishes an arity hierarchy for graph queries for quantifier-free update programs under insertions. The proof of the lower bound uses upper and lower bounds for Ramsey numbers.
The k-clique query can be maintained under edge inserts by a quantifier-free update program of arity k − 1 but not of arity k − 2 REF .
The Dynamic Descriptive Complexity of k-Clique
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Mathematics Computer Science