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S. Bova, R. Ganian, S. Szeider:
"Quantified conjunctive queries on partially ordered sets";
Theoretical Computer Science, 618 (2016), 72 - 84.

We study the computational problem of checking whether a quantified conjunctive query (a first-order sentence built using only conjunction as Boolean connective) is true in a finite poset (a reflexive, antisymmetric, and transitive directed graph). We prove that the problem is already NP-hard on a certain fixed poset, and investigate structural properties of posets yielding fixed-parameter tractability when the problem is parameterized by the query. Our main algorithmic result is that model checking quantified conjunctive queries on posets is fixed-parameter tractable when parameterized by the sentence and the width of the poset (the maximum size of a subset of pairwise incomparable elements). We complement our algorithmic result by complexity results with respect to classes of finite posets in a hierarchy of natural poset invariants, establishing its tightness in this sense.

http://dx.doi.org/10.1016/j.tcs.2016.01.010

http://publik.tuwien.ac.at/files/publik_253429.pdf