Name of the class of graphs obtained by deleting $\mathcal{Q}_d$ from $\mathcal{Q}_n$ Let $\mathcal{Q}_n$ denote the $n$-cube graph. I would like to know if there is a name for the class of graphs obtained by deleting a ${\bf single}$ arbitrary copy of $\mathcal{Q}_d$ from $\mathcal{Q}_n$, where $1 \leq d &lt; n$. For example, when $d = 1$, this would involve removing two vertices that are connected by an edge, when $d = 2$, this would involve removing a square (i.e $4$ vertices of $\mathcal{Q}_2$) from $Q_n$, and so forth. Do these types of graphs have a name/have been studied previously?

Edit: I know that they are partial cubes - https://en.wikipedia.org/wiki/Partial_cube - i.e an isometric subgraph of the hypercube, but I suspect something more specific can be said.
It look's like horn in simplicial sets / cubical sets (some explanations here). (N.B. It is not about graphs exactly. It is much wider and uses constructions of the nature @user340082710 asks about.)

Some recent works with them seem to be tightly connected with contemporary well-computated foundations of mathematics (homotopy type theory, cubical type theory). Look these references and notes to obtain more information.
It look's like horn in simplicial sets / cubical sets (some explanations here). (N.B. It is not about graphs exactly. It is much wider and uses constructions of the nature @user340082710 asks about.)

Some recent works with them seem to be tightly connected with contemporary well-computated foundations of mathematics (homotopy type theory, cubical type theory). Look these references and notes to obtain more information.