Finite invariant sets for a family of endomorphisms in compact graphs
Let \(G\) be an infinite graph. For \(A\subseteq V(G)\) the set of vertices of \(G\) which either belong to or are infinitely linked to \(A\) is denoted by \(\overline{A}\). By \textit{N. Polat} [Discrete Math. 158, No. 1-3, 211-221 (1996; Zbl 0859.05049)], \(A\rightarrow\overline A\) is the closure operator of a topology on \(V(G)\). It is also proved in the above cited paper that \(V(G)\) is compact if \(G\) is connected and rayless. This result is generalized in the present paper, showing that the vertex set of a connected graph \(G\) is compact if and only if each end of \(G\) is dominated. Another result states that if \(G\) is a connected \(T_{\aleph_0}\)-free graph whose ends are all dominated, then there exists a finite non-empty set of vertices of \(G\) that is invariant under every locally finite self-contraction of \(G\). Several consequences of this theorem are also proved.
scientific article; zbMATH DE number 1225620