SIAM Journal on Discrete Mathematics, Volume 35, Issue 1, Page 55-90, January 2021.
This paper deals with the characterization and the recognition of graph classes. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately, this strategy rarely leads to a very efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques that use some ordering of the nodes, such as the one given by a traversal. We specifically study graphs that have an ordering avoiding some ordered structures. More precisely, we consider structures that we call patterns on three nodes, and we study the complexity of recognizing the classes associated with such patterns. In this domain, there are three key previous works. Independently Skrien [J. Graph Theory, 6 (1982), pp. 309--316] and Damashke [Forbidden ordered subgraphs, in Topics in Combinatorics and Graph Theory, Physica-Verlag HD, 1990, pp. 219--229] noted that several graph classes, such as chordal, bipartite, interval, and comparability graphs, have a characterization in terms of forbidden patterns. On the algorithmic side, Hell, Mohar, and Rafiey [Ordering without forbidden patterns, in Algorithms--ESA 2014, Springer, 2014, pp. 554--565] proved that any class defined by a set of forbidden patterns on three nodes can be recognized in time $O(n^3)$ by using an algorithm based on an extension of 2-SAT. We improve on these two lines of works by systematically characterizing all the classes defined by sets of forbidden patterns (on three nodes) and proving that among the 22 different classes (up to complement) that we find, 20 can actually be recognized in linear time. Beyond these results, we consider that this type of characterization is very useful from an algorithmic perspective, leads to a rich structure of classes, and generates many algorithmic and structural open questions worth investigating.

中文翻译：

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三个顶点上的图类和禁止模式

SIAM 离散数学杂志，第 35 卷，第 1 期，第 55-90 页，2021 年 1 月。
本文涉及图类的表征和识别。表征图类的一种流行方法是列出一组最小的禁止诱导子图。不幸的是，这种策略很少导致非常有效的识别算法。另一方面，许多图类可以通过使用节点的某种排序的技术有效地识别，例如通过遍历给出的排序。我们专门研究具有避免某些有序结构的排序的图。更准确地说，我们考虑在三个节点上称为模式的结构，并研究识别与此类模式相关的类的复杂性。在这个领域，有三个关键的先前工作。独立的 Skrien [J. 图论, 6 (1982), pp. 309--316] 和 Damashke [禁止有序子图，在 Topics in Combinatorics and Graph Theory, Physica-Verlag HD, 1990, pp. 219--229] 指出几个图类，如弦图、二分图、区间图和可比性图，在禁止模式方面具有特征。在算法方面，Hell、Mohar 和 Rafiey [Ordering without forbidden patterns, in Algorithms--ESA 2014, Springer, 2014, pp. 554--565] 证明了由三个节点上的一组禁止模式定义的任何类都可以通过使用基于 2-SAT 扩展的算法，及时识别 $O(n^3)$。我们通过系统地表征由一组禁止模式（在三个节点上）定义的所有类，并证明在我们发现的 22 个不同的类（最多补充）中，实际上可以在线性时间内识别 20 个，我们改进了这两行工作. 除了这些结果，