  This paper establishes alternative characterizations of very expressive
classes of existential rule sets with decidable query entailment. We consider
the notable class of greedy bounded-treewidth sets (gbts) and a new,
generalized variant, called weakly gbts (wgbts). Revisiting and building on the
notion of derivation graphs, we define (weakly) cycle-free derivation graph
sets ((w)cdgs) and employ elaborate proof-theoretic arguments to obtain that
gbts and cdgs coincide, as do wgbts and wcdgs. These novel characterizations
advance our analytic proof-theoretic understanding of existential rules and
will likely be instrumental in practice.
