Automorphisms of the sphere complex of an infinite graph

For a locally finite, connected graph \Gamma , let \operatorname{Map}(\Gamma) denote the group of proper homotopy equivalences of \Gamma up to proper homotopy. Excluding sporadic cases, we show \operatorname{Aut}(\mathcal{S}(M_{\Gamma})) \cong \operatorname{Map}(\Gamma) , where \mathcal{S}(M_{\Gamma}) is the sphere complex of the doubled handlebody M_{\Gamma} associated to \Gamma . We also construct an exhaustion of \mathcal{S}(M_{\Gamma}) by finite strongly rigid sets when \Gamma has finite rank and finitely many rays, and an appropriate generalization otherwise.