  We consider existential rules (aka Datalog+) as a formalism for specifying
ontologies. In recent years, many classes of existential rules have been
exhibited for which conjunctive query (CQ) entailment is decidable. However,
most of these classes cannot express transitivity of binary relations, a
frequently used modelling construct. In this paper, we address the issue of
whether transitivity can be safely combined with decidable classes of
existential rules.
  First, we prove that transitivity is incompatible with one of the simplest
decidable classes, namely aGRD (acyclic graph of rule dependencies), which
clarifies the landscape of `finite expansion sets' of rules.
  Second, we show that transitivity can be safely added to linear rules (a
subclass of guarded rules, which generalizes the description logic DL-Lite-R)
in the case of atomic CQs, and also for general CQs if we place a minor
syntactic restriction on the rule set. This is shown by means of a novel query
rewriting algorithm that is specially tailored to handle transitivity rules.
  Third, for the identified decidable cases, we pinpoint the combined and data
complexities of query entailment.
