  We study the boundedness problem for unions of conjunctive regular path
queries with inverses (UC2RPQs). This is the problem of, given a UC2RPQ,
checking whether it is equivalent to a union of conjunctive queries (UCQ). We
show the problem to be ExpSpace-complete, thus coinciding with the complexity
of containment for UC2RPQs. As a corollary, when a UC2RPQ is bounded, it is
equivalent to a UCQ of at most triple-exponential size, and in fact we show
that this bound is optimal. We also study better behaved classes of UC2RPQs,
namely acyclic UC2RPQs of bounded thickness, and strongly connected UCRPQs,
whose boundedness problem are, respectively, PSpace-complete and
$\Pi^p_2$-complete. Most upper bounds exploit results on limitedness for
distance automata, in particular extending the model with alternation and
two-wayness, which may be of independent interest.
