TITLE: Is there a name for this "stack" of graphs?
QUESTION [5 upvotes]: Let $G_1,\ldots,G_m$ be a sequence of graphs, all having the same number $n$ of vertices. For each pair $(G_i, G_{i+1})$ we add $n$ edges that connect the vertices of $G_i$ and $G_{i+1}$ bijectively. My question: Is there an established name for this "stack" of graphs?

REPLY [3 votes]: In the case that you only have two graphs, and they are the same, say $G$, then the edges between the two copies of $G$ can be described by a permutation, and your graphs are precisely the permutation graphs defined by Chartrand and Harary in their 1967 paper "Planar permutation graphs" [Ann. Inst. H. Poincaré Sect. B (N.S.), 3, pp. 433–438]. A cute example of such a "permutation graph" is the Petersen graph, which we can obtain with $G=C_5$.
(I cannot stress strongly enough that this is not the standard notion of permutation graphs!)