For a graph class $\mathcal{G}$, we define the $\mathcal{G}$-modular cardinality of a graph $G$ as the minimum size of a vertex partition of $G$ into modules that each induces a graph in $\mathcal{G}$. This generalizes other module-based graph parameters such as neighborhood diversity and iterated type partition. Moreover, if $\mathcal{G}$ has bounded modular-width, the W[1]-hardness of a problem in $\mathcal{G}$-modular cardinality implies hardness on modular-width, clique-width, and other related parameters. On the other hand, fixed-parameter tractable (FPT) algorithms in $\mathcal{G}$-modular cardinality may provide new ideas for algorithms using such parameters. Several FPT algorithms based on modular partitions compute a solution table in each module, then combine each table into a global solution. This works well when each table has a succinct representation, but as we argue, when no such representation exists, the problem is typically W[1]-hard. We illustrate these ideas on the generic $(\alpha, \beta)$-domination problem, which asks for a set of vertices that contains at least a fraction $\alpha$ of the adjacent vertices of each unchosen vertex, plus some (possibly negative) amount $\beta$. This generalizes known domination problems such as Bounded Degree Deletion, $k$-Domination, and $\alpha$-Domination. We show that for graph classes $\mathcal{G}$ that require arbitrarily large solution tables, these problems are W[1]-hard in the $\mathcal{G}$-modular cardinality, whereas they are fixed-parameter tractable when they admit succinct solution tables. This leads to several new positive and negative results for many domination problems parameterized by known and novel structural graph parameters such as clique-width, modular-width, and $cluster$-modular cardinality.

翻译：对于图类$\mathcal{G}$，我们定义图$G$的$\mathcal{G}$-模基数为使$G$的顶点划分成每个诱导子图均属于$\mathcal{G}$的模所需的最小模数。该参数推广了诸如邻域多样性和迭代类型划分等基于模的图参数。此外，若$\mathcal{G}$具有有界模宽度，则问题在$\mathcal{G}$-模基数下的W[1]-困难性意味着该问题在模宽度、团宽度及其他相关参数下也具有困难性。另一方面，基于$\mathcal{G}$-模基数的固定参数可解算法或可为使用此类参数的算法提供新思路。基于模划分的多项FPT算法在每个模中计算解表，随后将各表合并为全局解。当每个表具有简洁表示时，该方法效果良好；然而我们论证指出，若不存在这种表示，问题通常为W[1]-困难。我们以一般化的$(\alpha, \beta)$-支配问题为例阐述上述思想——该问题要求寻找顶点集，使得每个未选顶点至少包含其邻接顶点中比例为$\alpha$（可能附加负值$\beta$）的顶点。这推广了有界度删除、$k$-支配和$\alpha$-支配等经典支配问题。我们证明：对于需要任意大解表的图类$\mathcal{G}$，此类问题在$\mathcal{G}$-模基数下为W[1]-困难；而当解表具有简洁表示时，问题为固定参数可解。这为许多以团宽度、模宽度和簇模基数为参数化的支配问题带来若干新的正反结果。