. . . "Abstract A caterpillar, H, is a tree containing a path, P, such that every vertex of H is either in P or adjacent to P. Given a graph G, a caterpillar tolerance representation of tolerance t on a caterpillar H, is a mapping of each vertex v ??? V(G) to a subtree Hv ??? H, such that uv ??? E( G) if and only if Hu and H v share at least t vertices. ^ Denote by cat[h,t] the set of all graphs for whic" .