Graphs of trigonal curves and rigid isotopies of singular real algebraic curves of bidegree $(4,3)$ on a hyperboloid

A rigid isotopy of real algebraic curves of a certain class is a path in the space of curves of this class. The paper’s study completes the rigid isotopic classification of nonsingular real algebraic curves of bidegree (4,3) on a hyperboloid, begun by the author in earlier works. There are given the missing proofs of the uniqueness of the connected components for 16 classes of real algebraic curves of bidegree (4,3) having a single node or a cusp. The main technical tools are graphs of real trigonal curves on Hirzebruch surfaces.