Dually vertex-oblique graphs Original Research Article

A vertex with neighbours of degrees image has vertex type image. 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 image. G is vertex-oblique iff image is; but G and image 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 image; they exist iff image or image, and for image 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.