  Given vertex valencies admissible for a self-dual polyhedral graph, we
describe an algorithm to explicitly construct such a polyhedron. Inputting in
the algorithm permutations of the degree sequence can give rise to
non-isomorphic graphs.
  As an application, we find as a function of $n\geq 3$ the minimal number of
vertices for a self-dual polyhedron with at least one vertex of degree $i$ for
each $3\leq i\leq n$, and construct such polyhedra. Moreover, we find a
construction for non-self-dual polyhedral graphs of minimal order with at least
one vertex of degree $i$ and at least one $i$-gonal face for each $3\leq i\leq
n$.
