Balogh, Bollobas and Weinreich showed that a parameter that has since been termed the distinguishing number can be used to identify a jump in the possible speeds of hereditary classes of graphs at the sequence of Bell numbers. We prove that every hereditary class that lies above the Bell numbers and has finite distinguishing number contains a boundary class for well-quasi-ordering. This means that any such hereditary class which in addition is defined by finitely many minimal forbidden induced subgraphs must contain an infinite antichain. As all hereditary classes below the Bell numbers are well-quasi-ordered, our results complete the answer to the question of well-quasi-ordering for hereditary classes with finite distinguishing number. We also show that the decision procedure of Atminas, Collins, Foniok and Lozin to decide the Bell number (and which now also decides well-quasi-ordering for classes of finite distinguishing number) has run time bounded by an explicit (quadruple exponential) function of the order of the largest minimal forbidden induced subgraph of the class.

中文翻译：

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井准序与有限区分数

Balogh、Bollobas 和 Weinreich 表明，一个后来被称为区分数的参数可用于识别按贝尔数序列的遗传图类的可能速度的跳跃。我们证明了位于贝尔数之上并具有有限区分数的每个遗传类都包含一个用于准排序的边界类。这意味着任何由有限多个最小禁止诱导子图定义的遗传类必须包含无限反链。由于贝尔数以下的所有遗传类都是准有序的，我们的结果完成了对具有有限区分数的遗传类的准有序问题的回答。我们还证明了 Atminas、Collins 的决策过程，