Weak geodesic topology and fixed finite subgraph theorems in infinite partial cubes II. Fixed subgraph properties and infinite treelike partial cubes

We prove that if a partial cube G contains no divergent rays, then: (i) there exists a non-empty finite convex subgraph of G which is fixed by every automorphism of G; (ii) every self-contraction of G fixes a non-empty finite isometric subgraph of G; (iii) for every commuting family F of self-contractions of G, there exists a non-empty finite isometric subgraph of G which is fixed by every element of F. Moreover, we show that those fixed subgraphs can be specified for some particular partial cubes, as is the case for median graphs and more generally for netlike partial cubes. We give new examples of such special graphs by extending the finite concept of treelike partial cubes [B. Brešar et al., Tree-like isometric subgraphs of hypercubes, Discussiones Mathematicae–Graph Theory 23 (2003), 227–240] to the infinite case, and by introducing what we call faithful-treelike, convex-treelike and strongly treelike partial cubes.