We study two graph parameters defined via tree decompositions: tree-independence number and induced matching treewidth. Both parameters are defined similarly as treewidth, but with respect to different measures of a tree decomposition $\mathcal{T}$ of a graph $G$: for tree-independence number, the measure is the maximum size of an independent set in $G$ included in some bag of $\mathcal{T}$, while for the induced matching treewidth, the measure is the maximum size of an induced matching in $G$ such that some bag of $\mathcal{T}$ contains at least one endpoint of every edge of the matching. While the induced matching treewidth of any graph is bounded from above by its tree-independence number, the family of complete bipartite graphs shows that small induced matching treewidth does not imply small tree-independence number. On the other hand, Abrishami, Briański, Czyżewska, McCarty, Milanič, Rzążewski, and Walczak~[SIAM Journal on Discrete Mathematics, 2025] showed that, if a fixed biclique $K_{t,t}$ is excluded as an induced subgraph, then the tree-independence number is bounded from above by some function of the induced matching treewidth. The function resulting from their proof is exponential even for fixed $t$, as it relies on multiple applications of Ramsey's theorem. In this note we show, using the Kövári-Sós-Turán theorem, that for any class of $K_{t,t}$-free graphs, the two parameters are in fact polynomially related.

翻译：我们研究通过树分解定义的两个图参数：树独立数与诱导匹配树宽。这两个参数的定义方式与树宽类似，但基于图G的树分解𝒯的不同度量：对于树独立数，其度量是𝒯中某个包（bag）所包含的G中最大独立集的大小；而对于诱导匹配树宽，其度量是G中一个诱导匹配的最大规模，且该匹配的每条边至少有一个端点包含在𝒯的某个包中。虽然任意图的诱导匹配树宽均由其树独立数上界约束，但完全二部图族表明小的诱导匹配树宽并不能推出小的树独立数。另一方面，Abrishami、Briański、Czyżewska、McCarty、Milanič、Rzążewski和Walczak~[SIAM Journal on Discrete Mathematics, 2025]证明，若排除固定二部团K_{t,t}作为诱导子图，则树独立数可由诱导匹配树宽的某个函数上界约束。其证明所导出的函数即使对固定t也是指数级的，因其依赖于拉姆齐定理的多次应用。本文中，我们利用Kövári-Sós-Turán定理证明，对于任何不含K_{t,t}的图类，这两个参数实际上存在多项式关联。