Faithful realizability of tropical curves

We study whether a given tropical curve Γ in ℝⁿ can be realized as the tropicalization of an algebraic curve whose non- Archimedean skeleton is faithfully represented by Γ . We give an affirmative answer to this question for a large class of tropical curves that includes all trivalent tropical curves, but also many tropical curves of higher valence. We then deduce that for every metric graph G with rational edge lengths, there exists a smooth algebraic curve in a toric variety whose analytification has skeleton G, and the corresponding tropicalization is faithful. Our approach is based on a combination of the theory of toric schemes over discrete valuation rings and logarithmically smooth deformation theory, expanding on a framework introduced by Nishinou and Siebert.