Maximal chains of isomorphic subgraphs of countable ultrahomogeneous graphs

For a countable ultrahomogeneous graph G = < G, rho > let P(G) denote the collection of sets A subset of G such that < A, rho boolean AND n [A](2)> congruent to G. The order types of maximal chains in the poset < P(G)boolean OR{empty set}, subset of > are characterized as: (I) the order types of compact sets of reals having the minimum non-isolated, if G is the Rado graph or the Henson graph H-n, for some n >= 3; (II) the order types of compact nowhere dense sets of reals having the minimum non-isolated, if G is the union of mu disjoint complete graphs of size nu, where mu nu = omega. (C) 2014 Elsevier Inc. All rights reserved.