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 ℵ0-regular tree (resp. or K1,1,ℵ0): (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 (resp. and which are interval-preserving)) 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.
