  A vertex with neighbours of degrees $d_1 \geq ... \geq d_r$ has {\em vertex
type} $(d_1, ..., d_r)$. A graph is {\em vertex-oblique} if each vertex has a
distinct vertex-type. While no graph can have distinct degrees, Schreyer,
Walther and Mel'nikov [Vertex oblique graphs, same proceedings] have
constructed infinite classes of {\em super vertex-oblique} graphs, where the
degree-types of $G$ are distinct even from the degree types of $\bar{G}$.
  $G$ is vertex oblique iff $\bar{G}$ is; but $G$ and $\bar{G}$ cannot be
isomorphic, since self-complementary graphs always have non-trivial
automorphisms. However, we show by construction that there are {\em dually
vertex-oblique graphs} of order $n$, where the vertex-type sequence of $G$ is
the same as that of $\bar{G}$; they exist iff $n \equiv 0$ or $1 \pmod 4, n
\geq 8$, and for $n \geq 12$ we can require them to be split graphs.
  We also show that a dually vertex-oblique graph and its complement are never
the unique pair of graphs that have a particular vertex-type sequence; but
there are infinitely many super vertex-oblique graphs whose vertex-type
sequence is unique.
