The Dynamic Descriptive Complexity of k-Clique

TL;DR

This paper investigates the dynamic descriptive complexity of maintaining k-clique queries in graphs, establishing an arity hierarchy for quantifier-free update programs under edge insertions.

Contribution

It demonstrates that k-clique can be maintained with arity k-1 but not with arity k-2, revealing a hierarchy in graph query complexity.

Findings

k-clique can be maintained with quantifier-free update programs of arity k-1

k-clique cannot be maintained with quantifier-free update programs of arity k-2

establishes an arity hierarchy for graph queries under insertions

Abstract

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.