It is natural in some contexts to extend word equations with regular or rational constraints. Such constraints restrict the space of solutions, by imposing some pairs of variables to be witnessed by a pair of words in a given regular or rational language. As of late, word equations with such constraints have played a prominent role in understanding the complexity of evaluation for graph query languages with path comparisons. These languages find applications, e.g., in provenance, route-finding, social networks, and the semantic web.
In this tutorial, we present several results generated in the interplay of word equations with rational constraints and graph querying. While word equations with rational constraints are, in general, undecidable, the focus on "practical" rational relations -- e.g., prefix, subsequence, or suffix -- that graph querying imposes force us to revisit such results and perform a more fine grained analysis of the complexity of such equations. On the other hand, we apply known techniques from word equations with constraints to obtain bounds for graph query evaluation that would have been difficult to obtain otherwise.