2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60054We prove several criteria for quasi-isometry between non-locally-finite graphs and their structure trees. Results of Möller in \cite{moeller92ends2} for locally finite and transitive graphs are generalized. We also give a criterion which describes quasi-isometry by how edge-ends are split up by the cuts of a structure tree.16 pagesCombinatoricsGroup Theory05C75; 05C25; 20B27text