  We study tropical line arrangements associated to a three-regular graph $G$
that we refer to as \emph{tropical graph curves}. Roughly speaking, the
tropical graph curve associated to $G$, whose genus is $g$, is an arrangement
of $2g-2$ lines in tropical projective space that contains $G$ (more precisely,
the topological space associated to $G$) as a deformation retract. We show the
existence of tropical graph curves when the underlying graph is a
three-regular, three-vertex-connected planar graph. Our method involves
explicitly constructing an arrangement of lines in projective space, i.e. a
graph curve whose tropicalisation yields the corresponding tropical graph curve
and in this case, solves a topological version of the tropical lifting problem
associated to canonically embedded graph curves. We also show that the set of
tropical graph curves that we construct are connected via certain local
operations. These local operations are inspired by Steinitz' theorem in
polytope theory.
