A framework that encodes seven fundamental properties of cellular graph embeddings, such as orientability, bipartiteness, directability, and 2-face-colorability, as the nonzero vectors of the binary space Z_2^3, so that the properties become the points of the Fano plane and their consistent combinations become its projective subspaces. The structure yields metatheorems governing which property sets can co-occur and characterizes when an embedding has a twisted dual with a given property via medial graphs and graph-encoded-map representations.
This paper introduces a 'Fano framework' unifying seven fundamental properties of cellular embeddings of graphs in compact surfaces, including orientability, bipartiteness, directability, 2-face-colo…