Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class ${\cal G}$ if they are so on the atoms (graphs with no clique cut-set) of ${\cal G}$. Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph $G$ is $H$-free if $H$ is not an induced subgraph of $G$, and it is $(H_1,H_2)$-free if it is both $H_1$-free and $H_2$-free. A class of $H$-free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for $(H_1,H_2)$-free graphs, as evidenced by one known example. We prove the existence of another such pair $(H_1,H_2)$ and classify the boundedness of clique-width on $(H_1,H_2)$-free atoms for all but 18 cases.

中文翻译：

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Clique-Width：利用原子的力量

许多 NP 完全图问题在有界集团宽度的图类上是多项式时间可解的。如果这些问题在 ${\cal G}$ 的原子（没有集团割集的图）上是这样，那么这些问题中的一些是多项式时间可解的，在遗传图类 ${\cal G}$ 上。因此，我们开始对遗传图类的原子团宽度的有界性进行系统研究。如果$H$ 不是$G$ 的诱导子图，则图$G$ 是$H$-free 的，如果它同时是$H_1$-free 和$H_2$ 的图则是$(H_1,H_2)$-free -自由。一类无 $H$ 的图具有有界团宽当且仅当其原子具有此属性。正如一个已知示例所证明的那样，对于无 $(H_1,H_2)$ 的图不再如此。我们证明了另一个这样的对 $(H_1,H_2)$ 的存在，并对除 $(H_1,H_2)$ 之外的所有原子的团宽度的有界性进行分类，除了 18 种情况。