We create a framework for hereditary graph classes $\mathcal{G}^\delta$ built on a two-dimensional grid of vertices and edge sets defined by a triple $\delta=\{\alpha,\beta,\gamma\}$ of objects that define edges between consecutive columns, edges between non-consecutive columns (called bonds), and edges within columns. This framework captures all previously proven minimal hereditary classes of graph of unbounded clique-width, and many new ones, although we do not claim this includes all such classes. We show that a graph class $\mathcal{G}^\delta$ has unbounded clique-width if and only if a certain parameter $\mathcal{N}^\delta$ is unbounded. We further show that $\mathcal{G}^\delta$ is minimal of unbounded clique-width (and, indeed, minimal of unbounded linear clique-width) if another parameter $\mathcal{M}^\beta$ is bounded, and also $\delta$ has defined recurrence characteristics. Both the parameters $\mathcal{N}^\delta$ and $\mathcal{M}^\beta$ are properties of a triple $\delta=(\alpha,\beta,\gamma)$, and measure the number of distinct neighbourhoods in certain auxiliary graphs. Throughout our work, we introduce new methods to the study of clique-width, including the use of Ramsey theory in arguments related to unboundedness, and explicit (linear) clique-width expressions for subclasses of minimal classes of unbounded clique-width.

翻译：我们构建了一个基于二维网格顶点集的遗传图类$\mathcal{G}^\delta$框架，其边集由三元组$\delta=\{\alpha,\beta,\gamma\}$定义，该三元组包含：相邻列之间的边、非相邻列之间的边（称为键合边）以及列内边。该框架涵盖了所有先前证明的具有无界团宽的极小遗传图类，并包含大量新类，尽管我们未声称其囊括所有此类图类。我们证明：图类$\mathcal{G}^\delta$具有无界团宽当且仅当特定参数$\mathcal{N}^\delta$无界。进一步研究表明：当另一参数$\mathcal{M}^\beta$有界，且$\delta$具有定义递归特性时，$\mathcal{G}^\delta$是无界团宽（实际上是无界线性团宽）的极小类。参数$\mathcal{N}^\delta$和$\mathcal{M}^\beta$均为三元组$\delta=(\alpha,\beta,\gamma)$的属性，用于度量特定辅助图中不同邻域的数量。本研究引入了团宽研究的新方法，包括在无界性论证中使用拉姆齐理论，以及为无界团宽极小类的子类构建显式（线性）团宽表达式。