  In our pursuit of generic criteria for decidable ontology-based querying, we
introduce 'finite-cliquewidth sets' (FCS) of existential rules, a
model-theoretically defined class of rule sets, inspired by the cliquewidth
measure from graph theory. By a generic argument, we show that FCS ensures
decidability of entailment for a sizable class of queries (dubbed 'DaMSOQs')
subsuming conjunctive queries (CQs). The FCS class properly generalizes the
class of finite-expansion sets (FES), and for signatures of arity at most 2,
the class of bounded-treewidth sets (BTS). For higher arities, BTS is only
indirectly subsumed by FCS by means of reification. Despite the generality of
FCS, we provide a rule set with decidable CQ entailment (by virtue of
first-order-rewritability) that falls outside FCS, thus demonstrating the
incomparability of FCS and the class of finite-unification sets (FUS). In spite
of this, we show that if we restrict ourselves to single-headed rule sets over
signatures of arity at most 2, then FCS subsumes FUS.
