Mim-width and sim-width are among the most powerful graph width parameters, with sim-width more powerful than mim-width, which is in turn more powerful than clique-width. While several $\mathsf{NP}$-hard graph problems become tractable for graph classes whose mim-width is bounded and quickly computable, no algorithmic applications of boundedness of sim-width are known. In [Kang et al., A width parameter useful for chordal and co-comparability graphs, Theoretical Computer Science, 704:1-17, 2017], it is asked whether \textsc{Independent Set} and \textsc{$3$-Colouring} are $\mathsf{NP}$-complete on graphs of sim-width at most $1$. We observe that, for each $k \in \mathbb{N}$, \textsc{List $k$-Colouring} is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. Moreover, we show that if the same holds for \textsc{Independent Set}, then \textsc{Independent $\mathcal{H}$-Packing} is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. This problem is a common generalisation of \textsc{Independent Set}, \textsc{Induced Matching}, \textsc{Dissociation Set} and \textsc{$k$-Separator}. We also make progress toward classifying the mim-width of $(H_1,H_2)$-free graphs in the case $H_1$ is complete or edgeless. Our results solve some open problems in [Brettell et al., Bounding the mim-width of hereditary graph classes, Journal of Graph Theory, 99(1):117-151, 2022].

翻译：Mim-width和sim-width是最强大的图宽度参数之一，其中sim-width比mim-width更强大，而mim-width又比clique-width更强大。尽管多个$\mathsf{NP}$-困难图问题对mim-width有界且可快速计算的图类是可处理的，但关于sim-width有界性的算法应用仍未见报道。在[Kang等人，一种适用于弦图和共可比图的宽度参数，Theoretical Computer Science, 704:1-17, 2017]中，作者提出如下问题：\textsc{独立集}问题和\textsc{$3$-着色}问题在sim-width至多为$1$的图上是否是$\mathsf{NP}$-完全的？我们观察到，对每个$k \in \mathbb{N}$，\textsc{列表$k$-着色}问题在sim-width有界且可快速计算的图类上是多项式时间可解的。此外，我们证明如果同样的结论对\textsc{独立集}成立，则\textsc{独立$\mathcal{H}$-打包}问题在sim-width有界且可快速计算的图类上也是多项式时间可解的。该问题是\textsc{独立集}、\textsc{诱导匹配}、\textsc{解离集}和\textsc{$k$-分隔符}的公共推广。我们还在$H_1$为完全图或无边的条件下，对$(H_1,H_2)$-自由图的mim-width分类问题取得了进展。我们的结果解决了[Brettell等人，有界遗传图类的mim-width，Journal of Graph Theory, 99(1):117-151, 2022]中的若干开放问题。