  Regular path queries (RPQs) are an essential component of graph query
languages. Such queries consider a regular expression r and a directed
edge-labeled graph G and search for paths in G for which the sequence of labels
is in the language of r. In order to avoid having to consider infinitely many
paths, some database engines restrict such paths to be trails, that is, they
only consider paths without repeated edges. In this paper we consider the
evaluation problem for RPQs under trail semantics, in the case where the
expression is fixed. We show that, in this setting, there exists a trichotomy.
More precisely, the complexity of RPQ evaluation divides the regular languages
into the finite languages, the class Ttract (for which the problem is
tractable), and the rest. Interestingly, the tractable class in the trichotomy
is larger than for the trichotomy for simple paths, discovered by Bagan,
Bonifati, and Groz [JCSS 2020]. In addition to this trichotomy result, we also
study characterizations of the tractable class, its expressivity, the
recognition problem, closure properties, and show how the decision problem can
be extended to the enumeration problem, which is relevant to practice.
