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 particular, if G is planar, we show that G is an induced subgraph of the strong product of a path and a graph of tree-width 39. 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乘积结构定理，以及相关的图乘积结构结果，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是有界最大度图Q（如路径）与有界树宽图M的强乘积的子图，则G是Q与树宽有界图M'的强乘积的诱导子图，其中M'的树宽至多为Q的最大度与M的树宽的指数函数。特别地，若G是平面图，我们证明G是路径与树宽为39的图的强乘积的诱导子图。在证明该结果的过程中，我们引入并研究了H-团宽度，这是一种新的单一结构度量，它捕捉了传统乘积结构的遗传类比（其中，非正式地说，强乘积的一个因子来自图类H，另一个因子具有有界团宽度）。