Polat, Norbert Invariant subgraph properties in pseudo-modular graphs. (English) Zbl 0945.05039 Discrete Math. 207, No. 1-3, 199-217 (1999). Author’s abstract: We prove different invariant subgraph properties for pseudo-modular graphs, and in particular for pseudo-median graphs and ball-Helly graphs. For example we prove that, for any connected interval-finite pseudo-modular graph \(G\) containing no isometric rays or subdivision of an \(\aleph_0\)-regular tree (or \(K_{1,1,\aleph_0}\), resp.): (i) there exists a finite set of vertices of \(G\) that is strictly invariant under every automorphism of \(G\); (ii) any commuting family of certain self-maps called d-faithful (resp. g-faithful) that preserve or collapse the edges (for example functions whose inverse image of each vertex is finite (and which are interval-preserving, resp.)) of \(G\) has a common strictly invariant finite set of vertices. In particular for pseudo-median (resp. ball-Helly) graphs, the invariant finite set of vertices generates an invariant finite regular pseudo-median subgraph (resp. complete subgraph). Moreover the second result holds for ball-Helly graphs, as well as for rayless pseudo-modular graphs, by considering any self-map that preserves or collapses the edges. Reviewer: M.E.Watkins (Syracuse) Cited in 6 Documents MSC: 05C38 Paths and cycles Keywords:isometric ray; self-contraction; invariant subgraph properties; pseudo-modular graphs; pseudo-median graphs; ball-Helly graphs PDFBibTeX XMLCite \textit{N. Polat}, Discrete Math. 207, No. 1--3, 199--217 (1999; Zbl 0945.05039) Full Text: DOI