Dually vertex-oblique graphs

A vertex with neighbours of degrees d1 ≥ · · · ≥ dr has vertex type (d1, . . . , dr). A graph is 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 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 ≡ 0 or 1 (mod 4), n ≥ 8, and for n ≥ 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 vertextype sequence; but there are infinitely many super vertex-oblique graphs whose vertex-type sequence is unique.