Finite invariant sets for a family of endomorphisms in compact graphs
type: Item
instance of: scholarly article
title: Finite invariant sets for a family of endomorphisms in compact graphs
zbMATH Open document ID: 0903.05047
DOI: 10.1023/A:1004371421758
author: Norbert Polat
published in: Periodica Mathematica Hungarica
publication date: 1998-11-22T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 05C38
zbMATH DE Number: 1225620
zbMATH Keywords: infinite graph
zbMATH Keywords: end
zbMATH Keywords: endomorphism
zbMATH Keywords: fixed point property
zbMATH Keywords: dominate
zbMATH Keywords: compact
MaRDI profile type: Publication
full work available at URL: a:1004371421758
OpenAlex ID: W1813092532
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description: scientific article; zbMATH DE number 1225620