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. While $(\mathrm{tw},\omega)$-bounded graph classes are known to enjoy some good algorithmic properties related to clique and coloring problems, it is an interesting open problem whether $(\mathrm{tw},\omega)$-boundedness also has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by means of a new min-max graph invariant 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 $G$. 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 will be given in the third paper of the series [Dallard, Milani\v{c}, and \v{S}torgel, Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure].

翻译：2020年,我们开始对图表类进行系统化研究,其中树线值只能大,因为存在一个巨大的球类,我们称之为$( mathrm{ tw},\ omega) 。 虽然已知美元( mathrm{ tw},\ omega) 受美元约束的图表类享有与球类和彩色问题相关的一些良好的算法属性, 但一个有趣的开放问题: $( mathrm{ tw},\ omega) 受树线性( 树线性) 是否对独立的赛类也有有用的算法影响。 我们通过与树分解有关的新的微数变形图来部分回答这个问题。 我们定义了一个树分解 $( mathal{ t) 的独立的独立数, 由某袋( $\ mathcal { t} 所引发的$G$( mal- delifality) 数字。 与所有直径直径的直径直径直径( ) 和直径直径直径直径直径直径) 的直径直径直径直径直径等结果, 。