We introduce H-clique-width, a new structural measure of graphs that aims to provide a hereditary analogue of the traditional graph product structure. The definition naturally generalises the ordinary clique-width concept. As a result, for a class H of graphs (such as the class of paths), the H-clique-width of a graph G equals the least integer t such that G is isomorphic to an induced subgraph of the strong product of a graph from H and a graph of clique-width t. We study basic properties of H-clique-width and compare it to other established structural parameters of graphs. Notably, we prove that the celebrated Planar graph product structure theorem by Dujmovic et al., and related graph product structure results, can all be formulated with the induced subgraph containment relation. In particular, every planar graph is isomorphic to an induced subgraph of the strong product of a path and a graph of tree-width 39.

翻译：我们引入H-团宽度，这是一种新的图结构度量，旨在为传统图乘积结构提供继承性类比。该定义自然推广了普通团宽度的概念。因此，对于图类H（例如路径类），图G的H-团宽度等于满足以下条件的最小整数t：G同构于来自H的某个图与团宽度为t的图的强乘积的诱导子图。我们研究了H-团宽度的基本性质，并将其与其他已建立的图结构参数进行了比较。值得注意的是，我们证明了Dujmovic等人著名的平面图乘积结构定理及相关图乘积结构结果，均可通过诱导子图包含关系来表述。特别地，每个平面图都同构于一条路径与树宽为39的图的强乘积的诱导子图。