Shifted Lévy's Dragon Curve and Directed Graph

It is known that every point on Lévy’s Dragon Curve admits a natural representation as a complex power series. We introduce a directed graph ${\mathcal{G}}_1$ which characterizes this representation. In this paper, we study the translation of the curve by $s=-1\mathrm{/}2+i\mathrm{/}2$. We identify another directed graph ${\mathcal{G}}_2$, that characterizes the translated curve and exhibits a revolving structure analogous to that of ${\mathcal{G}}_1$.