Paths in graphs: bounded geometry and property A

We expose a class of discrete metric spaces, for which bounded geometry is equivalent to the property A of G. Yu. This class includes the coarse disjoint union of $(\mathbb{Z}\mathrm{/}2\mathbb{Z}{)}^n$, $n\in \mathbb{N}$, and consists of spaces of simple paths in a class of graphs that includes cactus graphs, with the metric defined as the number of edges in the symmetric difference of the paths. We also show that if a space in this class does not have bounded geometry then it contains a subspace of bounded geometry without property A.