Evaluate FOC₁ on well-behaved dense classes

Determine whether expressions of the first-order logic with counting fragment FOC₁ can be evaluated efficiently on well-behaved dense graph classes.

Background

The paper establishes almost-linear-time preprocessing and constant-time query answering for first-order counting on effectively nowhere dense classes, and extends related results to clique-guarded first-order logic with counting (cgFOC). It contrasts these results with hardness phenomena for unrestricted FOC and for certain extensions of cgFOC on simple sparse and dense classes.

Against this background, the authors identify efficient evaluation of FOC₁ expressions on well-behaved dense classes as an unresolved direction. FOC₁ is a tractable fragment of FOC on effectively nowhere dense classes, but its algorithmic status on suitable dense classes is not settled here.