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.

翻译：图嵌入，特别是作为网格的子图嵌入，是超大规模集成电路设计与图绘制领域的经典课题。本文研究了将图$G$嵌入宿主图的相关复杂度问题，其中宿主图为路径$P$与满足特定性质（如较小树宽、路径宽或树深度）的图$H$的强乘积。我们证明，即使对$G$和$H$施加诸多限制条件，该问题仍属于NP难问题。特别地，即使对于小路径宽的树，计算行路径宽与行树深度仍为NP难问题；而即便对于串并联图，计算行树宽仍属NP难问题。