Take the unit $d$-cube with vertices $\{0,1\}^d$, and restrict to the vertices that lie between (or within one of) a pair of parallel hyperplanes. These vertices form a graph whose edges are the edges of the cube, i.e. there is an edge joining two vertices if they differ in one coordinate.
For example, if $d=3$ and the bounding hyperplanes are $x+2y+2z=1$ and $x+2y+2z=3$, then the graph is $P_5$ embedded as $(0,1,0)–(1,1,0)–(1,0,0)–(1,0,1)–(0,0,1)$.
Is anything known about such graphs? In particular, is anything known about their diameter (or the diameter of their connected components, for the disconnected ones)? Is there such a graph that has a connected component whose diameter exceeds $2d$, say?