In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call $(\mathrm{tw},\omega )$-bounded. The family of $(\mathrm{tw},\omega )$-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size. While Chaplick and Zeman showed in 2017 that $(\mathrm{tw},\omega )$-bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, it is an interesting open problem to which extent $(\mathrm{tw},\omega )$-boundedness has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by identifying a sufficient condition for $(\mathrm{tw},\omega )$-bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent Packing problem and, as a consequence, for the weighted variants of the Independent Set and Induced Matching problems.

Our approach is based on a new min-max graph parameter related to tree decompositions. We define the independence number of a tree decomposition $\mathcal{T}$ of a graph as the maximum independence number over all subgraphs of G induced by some bag of $\mathcal{T}$. The tree-independence number of a graph G is then defined as the minimum independence number over all tree decompositions of G. Boundedness of the tree-independence number is a refinement of $(\mathrm{tw},\omega )$-boundedness that is still general enough to hold for all the aforementioned families of graph classes. Generalizing a result on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is given together with a tree decomposition with bounded independence number, then the Maximum Weight Independent Packing problem can be solved in polynomial time. Applications of our general algorithmic result to specific graph classes are given in the third paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure].

2020年，我们启动了对图类的系统研究，其中树宽只能很大，因为存在大派系，我们称之为$（\mathrm{TW},\omega ）$- 有界。的家人$（\mathrm{TW},\omega ）$有界图类为各种不同的图类系列提供了统一的框架，包括有界树宽的图类、有界独立数的图类、有界树宽的图的连通子图的交集图以及其中所有最小分隔符的大小是有限的。虽然查普利克和泽曼在 2017 年表明$（\mathrm{TW},\omega ）$有界图类享有一些与派系和着色问题相关的良好算法属性，在某种程度上这是一个有趣的开放问题$（\mathrm{TW},\omega ）$-有界性对于与独立集相关的问题具有有用的算法含义。我们通过确定一个充分条件来提供这个问题的部分答案$（\mathrm{TW},\omega ）$有界图类，允许使用多项式时间算法来解决最大权重独立包装问题，从而解决独立集和诱导匹配问题的加权变体。

我们的方法基于与树分解相关的新的最小-最大图参数。我们定义树分解的独立数$\mathcal{时间}$一个图的最大独立数，由某些包引起的G的所有子图$\mathcal{时间}$。然后，图G的树独立数被定义为G的所有树分解的最小独立数。树独立数的有界性是对$（\mathrm{TW},\omega ）$-有界性仍然足够普遍，足以适用于所有上述图类系列。概括 2006 年 Cameron 和 Hell 提出的弦图结果，我们表明，如果给出一个图以及具有有限独立数的树分解，则可以在多项式时间内解决最大权重独立包装问题。该系列的第三篇论文给出了我们的通用算法结果在特定图类中的应用[Dallard、Milanič 和 Štorgel，树宽与团数。三．具有禁止结构的图的树独立数]。