  We give a framework to study the connectedness of the set of zeros of power
series with coefficients in a finite subset $G\subset \mathbb{C}$. We prove
that the set of zeros in the unit disk is connected and locally connected if
some graph on the set $G$ of coefficients is connected. Furthermore, we apply
this result to the study of the Mandelbrot set $\mathcal{M}_n$ for fractal
$n$-gons. We prove that $\mathcal{M}_n$ is connected and locally connected for
any $n$.
