Reduced parameters [BKW, JCTB '26; BKRT, SODA '22] are defined via contraction sequences. Based on this framework, we introduce the reduced component max-leaf, denoted by $\operatorname{cml}^\downarrow$, where component max-leaf is the maximum number of leaves in any spanning tree of any connected component. Reduced component max-leaf is strictly sandwiched between clique-width and reduced bandwidth, it is bounded in unit interval graphs, and unbounded in planar graphs. We design polynomial-time algorithms for problems such as \textsc{Maximum Induced $d$-Regular Subgraph} and \textsc{Induced Disjoint Paths} in graphs given with a contraction sequence witnessing low $\operatorname{cml}^\downarrow$, unifying and extending tractability results for classes of bounded clique-width and unit interval graphs. We get the following collapses in sparse classes of bounded $\operatorname{cml}^\downarrow$: bounded maximum degree implies bounded treewidth, whereas $K_{t,t}$-subgraph-freeness implies strongly sublinear treewidth; we show the latter, more generally, for classes of bounded reduced cutwidth. We establish the former result by showing that graphs with bounded $\operatorname{cml}^\downarrow$ admit balanced separators dominated by a bounded number of vertices. We then showcase an application of the reduced parameters to establishing non-transducibility results. We prove that for most reduced parameters $p^\downarrow$ (including reduced bandwidth), the family of classes of bounded $p^\downarrow$ is closed under first-order transductions. We then answer a question of [BKW '26] by showing that the 3-dimensional grids have unbounded reduced bandwidth. As the class of planar graphs (or any class of bounded genus) has bounded reduced bandwidth [BKW '26], this reproves a recent result [GPP, LICS '25] that planar graphs do not first-order transduce the 3-dimensional grids.

翻译：约化参数[BKW, JCTB '26; BKRT, SODA '22]通过收缩序列定义。基于此框架，我们引入约化分量最大叶（记作$\operatorname{cml}^\downarrow$），其中分量最大叶是任意连通分量的任意生成树中叶节点的最大数量。约化分量最大叶严格介于团宽度与约化带宽之间：它在单位区间图中有限，在平面图中无界。对于给定收缩序列（见证低$\operatorname{cml}^\downarrow$）的图，我们为\textsc{极大诱导$d$-正则子图}和\textsc{诱导不相交路径}等问题设计了多项式时间算法，统一并扩展了有界团宽度和图单位区间图的可解性结果。在稀疏图中，我们得到以下有界$\operatorname{cml}^\downarrow$的坍缩性质：有界最大度蕴含有界树宽，而$K_{t,t}$-子图自由性蕴含强次线性树宽；更一般地，我们证明后者对约化切割宽有界的类也成立。通过证明有界$\operatorname{cml}^\downarrow$的图具有由有界个顶点主导的平衡分离器，我们建立了前一个结果。随后，我们展示了约化参数在不可传递性结果中的应用。我们证明对大多数约化参数$p^\downarrow$（包括约化带宽），有界$p^\downarrow$的类族在一阶传递下封闭。接着，我们通过证明三维网格具有无界的约化带宽，回答了[BKW '26]中的一个问题。由于平面图类（或任意有界亏格类）具有有界约化带宽[BKW '26]，这重新证明了近期结果[GPP, LICS '25]：平面图不能一阶传递三维网格。