Dynamic Distances in Hyperbolic Graphs

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.