Abstract
A cubical polytope is a convex polytope all of whose facets are combinatorial cubes. A d-polytope P is called almost simple if, in the graph of P, each vertex of P is d-valent or ( d + 1)-valent. We give a complete enumeration of all the almost simple cubical d-polytopes for d ⩾ 4, which is even valid for almost simple cubical ( d − 1)-spheres. This provides a complete enumeration of all the cubical d-polytopes with up to 2 d+1 vertices for d ⩾ 4. With a single exception, these are precisely the d-polytopes which can be embedded into a ( d + 1)-cube.
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