  We study whether a given tropical curve $\Gamma$ in $\mathbb{R}^n$ can be
realized as the tropicalization of an algebraic curve whose non-archimedean
skeleton is faithfully represented by $\Gamma$. 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.
