Two series of edge-$4$-critical Grötzsch–Sachs graphs generated by four curves in the plane

Let $G$ be a 4-regular planar graph and suppose that $G$ has a cycle decomposition $S$ (i.e., each edge of $G$ is in exactly one cycle of the decomposition) with every pair of adjacent edges on a face always in different cycles of $S$. Such a graph $G$ arises as a superposition of simple closed curves in the plane with tangencies disallowed. Graphs of this class are called Grötzsch–Sachs graphs. Two infinite families of edge-$4$-critical Grötzsch–Sachs graphs generated by four curves in the plane have been announced in [4]. In this paper, we present a complete proof of this result.