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Let $\mathcal{G}$ be the set of locally finite, connected rooted graphs $(G,v)$ up to isomorphism $\cong$.

Denote by $[G,v]_r$ the sub-graph of $(G,v)$ induced by the vertices at distance $\leq r$ from the vertex $v$ (i.e. conncted to $v$ by a path of lenght $\leq r$ in the graph $G$).

Thus it's possible to define a metric on the space $\mathcal{G}$ : $d( (G,v) , (H,u) ) = 1/\sup\{ r\geq1 \,\textrm{ s.t. }\, [G,v]_{r-1} \cong [H,u]_{r-1}\} $

My question: is the metric space $(\mathcal{G},d)$ $\sigma$-compact? I know it is complete and separable, can these facts help?

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