Stable graphs for a family of endomorphisms

We prove several fixed subgraph properties. In particular it is shown that if is a commuting family of contractions of a connected graph G without infinite path and infinite interval, then there exists a nonempty finite subgraph F which is invariant under any element of . In particular this subgraph F is a simplex if G is moreover a strongly dismantlable graph or a ball-Helly graph without infinite block, or if it is chordal. This implies that for any commuting family of contractions of a tree without infinite path, there is a common fixed vertex or a common fixed edge.