Treewidth is an important graph invariant, relevant for both structural and algorithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking induced subgraphs in which this condition is also sufficient, which we call $(tw,\omega)$-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and $k$-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, induced topological minor, and induced subgraph relations. For each of them, we give a complete characterization of the graphs $H$ for which the class of graphs excluding $H$ is $(tw,\omega)$-bounded. Our results yield an infinite family of $\chi$-bounded induced-minor-closed graph classes and imply that the class of $1$-perfectly orientable graphs is $(tw,\omega)$-bounded, leading to linear-time algorithms for $k$-coloring $1$-perfectly orientable graphs for every fixed~$k$. This answers a question of Bre\v sar, Hartinger, Kos, and Milani{\v c} from 2018 and one of Beisegel, Chudnovsky, Gurvich, Milani{\v c}, and Servatius from 2019, respectively. We also reveal some further algorithmic implications of $(tw,\omega)$-boundedness related to list $k$-coloring and clique problems. In addition, we propose a question about the complexity of the maximum weight independent set problem in $(tw,\omega)$-bounded graph classes and prove that the problem is polynomial-time solvable in every class of graphs excluding a fixed star as an induced minor.
翻译:树枝是一条重要的变数图, 与结构和算法上的原因相关。 图形类中绑定树枝的一个必要条件就是没有大硬度。 我们研究以诱导的子图类中封闭的, 这个条件也足够, 我们称之为$( tw,\ omega) 。 这些图表类中存在与 clui 和 $- log- log 问题相关的有用的算法应用。 我们考虑六个众所周知的图表封存关系: 小的、 地表小的、 Subgraph、 诱导的表层小和 导致的子座关系。 对于其中的每一个, 我们研究的图表类别中不包括$( tw,\ omga) $( omega) 。 这些图表中的最大组是 $( poliver- ormai) 问题, 也就是从 $( tw, omega) lidal- ligal- lexal- legal- roupal- commatial 答案中, 20美元 和 美元- 美元- 美元- 美元- 美元- 美元- tal- tal- kal- dalmaxlick- tal- tal- tal- tal- tal- tal- tal- tald- tal- 。