  Graph databases in many applications---semantic web, transport or biological
networks among others---are not only large, but also frequently modified.
Evaluating graph queries in this dynamic context is a challenging task, as
those queries often combine first-order and navigational features.
  Motivated by recent results on maintaining dynamic reachability, we study the
dynamic evaluation of traditional query languages for graphs in the descriptive
complexity framework. Our focus is on maintaining regular path queries, and
extensions thereof, by first-order formulas. In particular we are interested in
path queries defined by non-regular languages and in extended conjunctive
regular path queries (which allow to compare labels of paths based on word
relations). Further we study the closely related problems of maintaining
distances in graphs and reachability in product graphs.
  In this preliminary study we obtain upper bounds for those problems in
restricted settings, such as undirected and acyclic graphs, or under insertions
only, and negative results regarding quantifier-free update formulas. In
addition we point out interesting directions for further research.
