Finite invariant sets in infinite graphs

AbstractThe closure of a set A of vertices of an infinite graph G is defined as the set of vertices of G which cannot be finitely separated from A. A subset A of V(G) is dispersed if it is finitely separated from any ray of G. It is shown that the closure of any dispersed set A of an infinite connected graph G contains a nonempty finite subset which is invariant under any automorphism of G stabilizing A. Therefore any infinite connected graph not containing a ray has a finite set of vertices which is invariant under any automorphism. The same also holds for connected graphs with rays, containing no end-respecting subdivision of the dyadic trec, provided that there are at least three (resp. two) ends of maximal order (resp. and a vertex which cannot be finitely separated from one of them)