We prove that the celebrated Planar Product Structure Theorem by Dujmovic et al, and also related graph product structure results, can be formulated with the induced subgraph containment relation. Precisely, we prove that if a graph G is a subgraph of the strong product of a graph Q of bounded maximum degree (such as a path) and a graph M of bounded tree-width, then G is an induced subgraph of the strong product of Q and a graph M' of bounded tree-width being at most exponential in the maximum degree of Q and the tree-width of M. In the course of proving this result, we introduce and study H-clique-width, a new single structural measure that captures a hereditary analogue of the traditional product structure (where, informally, the strong product has one factor from the graph class H and one factor of bounded clique-width).

翻译：我们证明了Dujmovic等人提出的著名的平面积结构定理，以及相关的图积结构结果，可以用诱导子图包含关系来表述。具体而言，我们证明：若图G是最大度有界图Q（如路径）与树宽有界图M的强积的子图，则G是Q与树宽有界图M'的强积的诱导子图，其中M'的树宽至多为Q的最大度与M的树宽的指数函数。在证明该结果的过程中，我们引入并研究了H-团宽度——一种新的单一结构度量，它捕捉了传统积结构的遗传类比（非正式地说，强积的一个因子来自图类H，另一个因子具有有界团宽度）。