Dually vertex-oblique graphs

A vertex with neighbours of degrees d1⩾⋯⩾drd1⩾⋯⩾dr has vertex type  (d1,…,dr)(d1,…,dr). A graph is vertex-oblique if each vertex has a distinct vertex type (no graph can have distinct degrees). Schreyer et al. [Vertex-oblique graphs, same proceedings] have constructed infinite classes of super vertex-oblique graphs, where the degree types of G   are distinct even from the degree types of G¯.G   is vertex-oblique iff G¯ is; but G   and G¯ cannot be isomorphic, since self-complementary graphs always have non-trivial automorphisms. However, we show by construction that there are dually vertex-oblique graphs of order n, where the vertex-type sequence of G   is the same as that of G¯; they exist iff n≡0n≡0 or 1(mod4),n⩾8, and for n⩾12n⩾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.