  We consider the following dynamic problem: given a fixed (small) template
graph with colored vertices C and a large graph with colored vertices G (whose
colors can be changed dynamically), how many mappings m are there from the
vertices of C to vertices of G in such a way that the colors agree, and the
distances between m(v) and m(w) have given values for every edge? We show that
this problem can be solved efficiently on triangulations of the hyperbolic
plane, as well as other Gromov hyperbolic graphs. For various template graphs
C, this result lets us efficiently solve various computational problems which
are relevant in applications, such as visualization of hierarchical data and
social network analysis.
