  In this work, we establish local well-posedness for the Korteweg-de Vries
model on a balanced star graph with a structure represented by semi-infinite
edges, by considering a boundary condition of $\delta$-type at the {unique}
graph-vertex. Also, we extend the linear instability result of Angulo and
Cavalcante (2021) to one of nonlinear instability. For the proof of local well
posedness theory the principal new ingredient is the utilization of the special
solutions by Faminskii in the context of half-lines. As far as we are aware,
this approach is being used for the first time in the context of star graphs
and can potentially be applied to other boundary classes. In the case of the
nonlinear instability result, the principal ingredients are the linearized
instability known result and the fact that data-to-solution map determined by
the local theory is at least of class $C^2$.
