Netlike partial cubes, IV: Fixed finite subgraph theorems

We prove that, if a netlike partial cube G (see [N. Polat, Netlike partial cubes I. General properties, Discrete Math. 307 (2007) 2704–2722]) contains no isometric rays, then there exists a convex cycle or a finite hypercube which is fixed by every automorphism of G . Furthermore we prove that every self-contraction (map which preserves or collapses the edges) of G fixes a convex cycle or a finite hypercube if and only if G contains no isometric rays. We also study the self-contractions of G which fix no finite set of vertices.