Finite invariant sets in infinite graphs (Q1815321)

For a graph \(G\) and \(A,B\subseteq V(G)\), an \(AB\)-path of \(G\) is an \(xy\)-path of \(G\) whose only vertices in \(A\cup B\) are \(x\) and \(y\), with \(x\in A\), \(y\in B\). For \(x\in V(G)\) and \(A\subseteq V(G)\), an \(xA\)-linking of \(G\) is a set of \(xA\)-paths of \(G\), which have pairwise only \(x\) in common. If there is an infinite \(xA\)-linking, then we say that \(x\) is infinitely linked to \(A\) in \(G\). For \(A\subseteq V(G)\) we denote by \(\overline A\) the set of vertices of \(G\) which belong to \(A\) or which are infinitely linked to \(A\) in \(G\). \(\text{Stab}(A)=\{\sigma\in \Aut(G)\mid \sigma(A)=A\}\). An infinite subset \(S\) of \(V(G)\) is concentrated in \(G\) if for all infinite subsets \(T\), \(U\) of \(S\) there is an infinite family of pairwise disjoint \(TU\)-paths in \(G\). A subset \(S\) of \(V(G)\) is dispersed if \(S\) has no concentrated subset. The main result of this paper is: Theorem 3.6. Let \(A\) be a nonempty dispersed set of vertices of a connected graph \(G\). Then there is a nonempty finite subset \(F\) of \(\overline A\) such that \(\text{Stab}(A)\subseteq \text{Stab}(F)\).