2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169957In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces $MG_{1,n}^v$, which parametrize the isometry classes of metric graphs of genus 1 with $n$ marks on vertices are homotopy equivalent to the spaces $TM_{1,n}$, which are the moduli spaces of tropical curves of genus 1 with $n$ marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.Topology and its Applications, In Press, Corrected Proof, Available online 3 August 2008Algebraic TopologyAlgebraic Geometry57xx (Primary) 14Mxx (Secondary)text