Fixed finite subgraph theorems in infinite weakly modular graphs Original Research Article

We prove different fixed subgraph properties for some infinite weakly modular graphs. In particular we prove that every self-contraction (map which preserves or collapses the edges) of a weakly median graph G fixes a non-empty finite regular weakly median subgraph of G if and only if G is connected and contains no infinite simplices and no isometric rays. We also prove some fixed finite simplex theorems for other weakly modular graphs such as Helly graphs, chordal graphs and bridged graphs.