We introduce a novel generalization of the notion of clique-width which aims to bridge the gap between classical hereditary width measures and the recently introduced graph product structure theory. Bounding the new H-clique-width, in the special case of H being the class of paths, is equivalent to admitting a hereditary (i.e., induced) product structure of a path times a graph of bounded clique-width. Furthermore, every graph admitting the usual (non-induced) product structure of a path times a graph of bounded tree-width, has bounded H-clique-width and, as a consequence, it admits the usual product structure in an induced way. We prove further basic properties of H-clique-width in general.

翻译：本文提出了一种新的团宽度概念的推广，旨在弥合经典遗传宽度度量与近期引入的图乘积结构理论之间的差距。在特殊情形下，当H为路径类时，限制新定义的H-团宽度等价于允许路径与有界团宽度图的遗传（即诱导）乘积结构。此外，任何允许路径与有界树宽度图的常规（非诱导）乘积结构的图，都具有有界H-团宽度，并因此能以诱导方式实现常规乘积结构。我们进一步证明了H-团宽度的一般基本性质。