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Let x be a vertex of a simple graph G. The vertex-type of x is the lexicographically ordered degree sequence of its neighbors. We call the graph G vertex-oblique if there are no two vertices in V(G) which are of the same vertex-type. We will show that the set of vertex-oblique graphs of arbitrary connectivity is infinite.

The paper considers polyhedral graphs (graphs of polyhedron or planar 3-connected graphs). A face $\alpha$ of size $k$ of a polyhedral graph is of type $<a_1,a_2,...,a_k>$ if the vertices incident with $\alpha$ in cyclic order have degrees $a_1,a_2,...,a_k$ and this sequence is lexicographically minimal. A polyhedral graph is oblique if it has no two faces of the same type. The number of polyhedral graphs with up to 12 vertices is found. Some additional properties of such graphs are also considered.