Plasticity in graph metric spaces and their hyperspaces

Abstract

A metric space is plastic if every bijective nonexpansive self-map is an isometry. We prove that the hyperspace of nonempty compact subsets of every connected, locally finite, regular graph, equipped with the Hausdorff metric associated with the path metric, is plastic. The proof identifies the sets with smallest unit balls as the nonempty subsets of adjacent-twin classes and shows that every nonexpansive bijection induces an automorphism of the quotient graph preserving the sizes of these classes. Singletons are preserved when there are no adjacent twins, but need not be preserved in general. We also prove that $\mathcal{K}(K\times G)$ is plastic for every compact connected metric space $K$ and every connected, locally finite, regular graph $G$, with the supremum metric on the product, and give a more general criterion for such products. Further results include a plastic metric space whose hyperspace is not plastic, plasticity of every tree, and plasticity of the hyperspace of a connected, locally finite graph with only finitely many vertices of minimum degree.