Take a polytope $P$ and $G(P)$ it's corresponding graph. And $H$ a subgraph of $G(P)$ which is the graph of a polytope $\tilde{H}$. Does this imply that $\tilde{H}$ is a face of $P$?
2026-03-28 08:09:27.1774685367
subgraph of the graph of a polytope
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No. Take a double pyramid over the cycle $C_4$. The subgraph induced by the cycle is the graph of a polytope, but is not a face.