In low-dimensional topology and geometric group theory, an exciting strategy is to associate to a compact manifold or a finitely generated group a connected graph: Witness Gromov's programme to study the large-scale geometry of groups (via their Cayley graphs), or Minsky et al's solution to Thurston's ELC, or Hatcher-Thurston's algorithm giving finite presentations of surface mapping class groups. The aim always is to elicit an underlying principal. These objects then take on a life of their own...