Moduli spaces of metric graphs of genus 1 with marks on vertices

In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces , which parametrize the isometry classes of metric graphs of genus 1 with n marks on vertices are homotopy equivalent to the spaces TM1,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.