In this paper we study the class of bi-arc digraphs, important from two seemingly unrelated perspectives. On the one hand, they are precisely the digraphs that admit certain polymorphisms of interest in the study of constraint satisfaction problems; on the other hand, they are a very broad generalization of interval graphs. Bi-arc digraphs is the class of digraphs that admit conservative semilattice polymorphisms. There is much interest in understanding structures that admit particular types of polymorphisms, and especially in their recognition algorithms. (Such problems are referred to as metaproblems.) Surprisingly, the class of bi-arc digraphs also describes the class of digraphs that admit certain other kinds of conservative polymorphisms. Thus solving the recognition problem for bi-arc digraphs solves the metaproblem for digraphs for several types of conservative polymorphisms. The complexity of the recognition problem for digraphs with conservative semilattice polymorphisms was an open problem, while it was known to be NP-complete for certain more complex relational structures. We complement our result by providing a complete dichotomy classification of which general relational structures have polynomial or NP-complete recognition problems for the existence of conservative semilattice polymorphisms. Bi-arc digraphs also generalizes the class of interval graphs; in fact it reduces to the class of interval graphs for symmetric and reflexive digraphs. It is much broader than interval graphs and includes other generalizations of interval graphs such as co-threshold tolerance graphs and adjusted interval digraphs. Yet, it is still a reasonable extension of interval graphs, in the sense that it keeps much of the appeal of interval graphs. Our main result is a forbidden obstruction characterization of, and a polynomial recognition for, the class of bi-arc digraphs.

中文翻译：

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双弧有向图和保守多态性

在本文中，我们研究了双弧有向图的类别，从两个看似无关的角度来看很重要。一方面，它们正是在约束满足问题的研究中承认某些感兴趣的多态性的有向图；另一方面，它们是区间图的非常广泛的概括。双弧有向图是一类承认保守半格多态性的有向图。人们对理解允许特定类型的多态性的结构很感兴趣，尤其是在它们的识别算法中。（此类问题被称为元问题。）令人惊讶的是，双弧有向图类还描述了承认某些其他类型的保守多态性的有向图类。因此，解决双弧有向图的识别问题就解决了几种保守多态性的有向图的元问题。具有保守半格多态性的有向图识别问题的复杂性是一个开放问题，而对于某些更复杂的关系结构，它被称为 NP 完全问题。我们通过提供完整的二分法分类来补充我们的结果，其中一般关系结构对于保守半格多态性的存在具有多项式或 NP 完全识别问题。双弧有向图也概括了区间图的类别；事实上，它归结为对称和自反有向图的区间图类。它比区间图更广泛，包括区间图的其他概括，例如共同阈值容差图和调整区间有向图。然而，它仍然是区间图的合理扩展，因为它保留了区间图的大部分吸引力。我们的主要结果是双弧有向图类的禁止障碍表征和多项式识别。