Real dihedral $p$-gonal Riemann surfaces

Riemann surfaces (and algebraic curves) have been comprehensively studied when they are regular (Galois) coverings of the Riemann sphere, but barely addressed in the general case of being non-regular coverings. In this article we deal with this less known case for a special type of non-regular $p$-coverings ($p$ prime greater than 2), those with monodromy group isomorphic to the dihedral group $D_p$, which we call dihedral $p$-gonal coverings (the particular case $p=3$ has been already studied by A. F. Costa and M. Izquierdo). We have focused on real algebraic curves (those that have a special anticonformal involution) and we study real dihedral $p$-gonal Riemann surfaces. We found out the restrictions, besides Harnack's theorem and generalizations, that apply to the possible topological types of real dihedral $p$-gonal Riemann surfaces.