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David Wood asked me this question in June 2017. It asks about finding a relationship between the page number of a graph and its layered treewidth. (A definition of layered treewidth can be found in Section 2 of this paper.)

Is the page number of every graph bounded a function of its layered treewidth?

The simplest case is graphs of layered treewidth 1, which can be visualized as follows: Take the product of a tree $T$ and a path of length $k$. The resulting graph naturally has some tree edges ($k$ copies of each edge in $T$) and non-tree edges. Next perform arbitrary edge contractions on the tree edges to obtain a graph $G$. Every graph of layered treewidth 1 is a subgraph of a graph that can be obtained this way, so it suffices to bound the page number of $G$.

The general case is probably far from trivial. It would imply, for example, that $k$-planar graphs have bounded page number. We were able to solve the special case where $G$ has small layered pathwidth.

Update (May 2019)

In view of the recent developments with layered $H$-partitions, it probably makes as much sense to ask this question about graphs have layered $H$-partitions of bounded layered width in which $H$ has bounded treewidth.

In fact, it might make the most sense to find a structural decomposition like $H$-partitions (but maybe using a DFS rather than a BFS tree).