Enumeration for FO Queries over Nowhere Dense Graphs

We consider the evaluation of first-order queries over classes databases that are nowhere dense . The notion was introduced by Nešetřil and Ossona de Mendez as a formalization “sparse” graphs generalizes many well-known graphs, such bounded degree, tree-width, or expansion. It has recently been shown Grohe, Kreutzer, Siebertz databases, sentences can be evaluated in pseudo-linear time (pseudo-linear means for all \( \epsilon \) there exists an algorithm working O(n^{1+\epsilon }) , where n is size database). For higher arities, we show any class set their solutions enumerated with constant delay after preprocessing. In same context, also preprocessing can, on input tuple, test whether it solution to query.