reference request – Name of the facet of a cone containing all edges except one

Let $ C subseteq mathbb R ^ n $ to be a polyhedral cone, thus generated by its edges ($ 1 $(three-dimensional faces) and $ F subseteq C $ a facet (codimension $ 1 $ face) containing all edges except $ e $. In particular, the map $ F times e stackrel {+} { to} C $ is an isomorphism of the cones.

Is there a name for this property of $ F $ or $ e $ (who are determined)?

Essentially, I consider the edges $ e $ like this as a kind of trivial, and would like to separate them to deal with the difficult part of $ C $. So maybe we could talk about a "main facet" $ F $ of $ C $and let the "core of $ C $"is the intersection of the" main facets ". Then the map $ core (C) times production _ { text {core edge} e} e stackrel {+} { to} C $ would be an isomorphism, and $ core (C) $ would have no basic edges.

There is a similar and familiar construction in the theory of simplicial complexes, where a "cone-top" $ v $ of $ Delta $ is a lying in each maximum face. You can safely remove all the vertices of the cones, and recreate them $ Delta $. Obviously, we do not want to steal this terminology directly and talk about "conical edges".