Regular path queries (RPQ) is a classical navigational query formalism for graph databases to specify constraints on labeled paths. Recently, RPQs have been extended by Libkin and Vrgo$\rm \check{c}$ to incorporate data value comparisons among different nodes on paths, called regular path queries with data (RDPQ). It has been shown that the evaluation problem of RDPQs is PSPACE-complete and NLOGSPACE-complete in data complexity. On the other hand, the containment problem of RDPQs is in general undecidable. In this paper, we propose a novel approach to extend regular path queries with data value comparisons, called rigid regular path queries with data (RRDPQ). The main ingredient of this approach is an automata model called nondeterministic rigid register automata (NRRA), in which the data value comparisons are \emph{rigid}, in the sense that if the data value in the current position $x$ is compared to a data value in some other position $y$, then by only using the labels (but not data values), the position $y$ can be uniquely determined from $x$. We show that NRRAs are robust in the sense that nondeterministic, deterministic and two-way variant of NRRAs, as well as an extension of regular expressions, are all of the same expressivity. We then argue that the expressive power of RDPQs are reasonable by demonstrating that for every graph database, there is a localized transformation of the graph database so that every RDPQ in the original graph database can be turned into an equivalent RRDPQ over the transformed one. Finally, we investigate the computational properties of RRDPQs and conjunctive RRDPQs (CRRDPQ). In particular, we show that the containment of CRRDPQs (and RRDPQs) can be decided in 2EXPSPACE.