Fixing subgraphs and Ulam's conjecture

A spanning subgraph U of a graph G belongs to the set $\text{J}$(G) of fixing subgraphs (see [5]) of G if every embedding of U into G can be extended to an automorphism of G. Clearly G ∈ $\text{J}$(G). G is free if …$\text{J}$(G)… = 1. We establish a connection between Ulam's conjecture and free graphs and continue with an investigation of free graphs.