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. Our study completes the rigid isotopic classification of nonsingular real algebraic curves of bidegree (4,3) on a hyperboloid, started by the author in his earlier works. 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 are given. The main technical tools are graphs of real trigonal curves on Hirzebruch surfaces.