  Answer Set Programming (ASP) is a generic problem modeling and solving
framework with a strong focus on knowledge representation and a rapid growth of
industrial applications. So far, the study of complexity resulted in
characterizing hardness and determining their sources, fine-grained insights in
the form of dichotomy-style results, as well as detailed parameterized
complexity landscapes. Unfortunately, for the well-known parameter treewidth
disjunctive programs require double-exponential runtime under reasonable
complexity assumptions. This quickly becomes out of reach. We deal with the
classification of structural parameters for disjunctive ASP on the program's
rule structure (incidence graph).
  First, we provide a polynomial kernel to obtain single-exponential runtime in
terms of vertex cover size, despite subset-minimization being not represented
in the program's structure. Then we turn our attention to strictly better
structural parameters between vertex cover size and treewidth. Here, we provide
double-exponential lower bounds for the most prominent parameters in that
range: treedepth, feedback vertex size, and cliquewidth. Based on this, we
argue that unfortunately our options beyond vertex cover size are limited. Our
results provide an in-depth hardness study, relying on a novel reduction from
normal to disjunctive programs, trading the increase of complexity for an
exponential parameter compression.
