We show that the problem of whether a query is equivalent to a query of tree-width k is decidable, for the class of Unions of Conjunctive Regular Path Queries with two-way navigation (UC2RPQs). A previous result by Barceló, Romero, and Vardi [3] has shown decidability for the case k = 1, and here we show that decidability in fact holds for any arbitrary k > 1. The algorithm is in 2ExpSpace, but for the restricted but practically relevant case where all regular expressions of the query are of the form a* or (a_1 + … + a_n) we show that the complexity of the problem drops to Π_2^p. We also investigate the related problem of approximating a UC2RPQ by queries of small tree-width. We exhibit an algorithm which, for any fixed number k, builds the maximal under-approximation of tree-width k of a UC2RPQ. The maximal under-approximation of tree-width k of a query q is a query q ′ of tree-width k which is contained in q in a maximal and unique way, that is, such that for every query q′′ of tree-width k, if q′′ is contained in q then q′′ is also contained in q′.