On the complexity of embedding in graph products

Graph embedding, especially as a subgraph of a grid, is an old topic in VLSI design and graph drawing. In this paper, we investigate related questions concerning the complexity of embedding a graph $G$ in a host graph that is the strong product of a path $P$ with a graph $H$ that satisfies some properties, such as having small treewidth, pathwidth or tree depth. We show that this is NP-hard, even under numerous restrictions on both $G$ and $H$. In particular, computing the row pathwidth and the row treedepth is NP-hard even for a tree of small pathwidth, while computing the row treewidth is NP-hard even for series-parallel graphs.