A graph database is a digraph whose arcs are labelled with symbols from a fixed alphabet. A regular graph pattern (RGP) is a digraph whose edges are labelled with regular expressions over the alphabet. RGPs model navigational queries for graph databases, more precisely, conjunctive regular path queries. A match of a navigational RGP query in the database is witnessed by a special navigational homomorphism of the RGP to the database. We study the complexity of deciding the existence of a homomorphism between two RGPs. Such homomorphisms model a strong type of containment between two navigational RGP queries. We show that this problem can be solved by an EXPTIME algorithm (while general query containment in this context is EXPSPACE-complete). We also study the problem for restricted RGPs over a unary alphabet, that arise from some applications like XPath, and prove that certain interesting cases are polynomial-time solvable.