A popular way to define or characterize graph classes is via forbidden subgraphs or forbidden minors. These characterizations play a key role in graph theory, but they rarely lead to efficient algorithms to recognize these classes. In contrast, many essential graph classes can be recognized efficiently thanks to characterizations of the following form: there must exist an ordering of the vertices such that some ordered pattern does not appear, where a pattern is basically an ordered subgraph. These pattern characterizations have been studied for decades, but there have been recent efforts to better understand them systematically. In this paper, we focus on a simple problem at the core of this topic: given an ordered graph of size $n$, how fast can we detect whether a fixed pattern of size $k$ is present? Following the literature on graph classes recognition, we first look for patterns that can be detected in linear time. We prove, among other results, that almost all patterns on three vertices (which capture many interesting classes, such as interval, chordal, split, bipartite, and comparability graphs) fall in this category. Then, in a finer-grained complexity perspective, we prove conditional lower bounds for this problem. In particular we show that for a large family of patterns on four vertices it is unlikely that subquadratic algorithm exist. Finally, we define a parameter for patterns, the merge-width, and prove that for patterns of merge-width $t$, one can solve the problem in $O(n^{ct})$ for some constant~$c$. As a corollary, we get that detecting outerplanar patterns and other classes of patterns can be done in time independent of the size of the pattern.

翻译：一种定义或刻画图类的流行方式是通过禁止子图或禁止 minors。这些刻画在图论中起着关键作用，但它们很少能带来识别这些图类的高效算法。相比之下，许多重要图类可以通过以下形式的刻画得到高效识别：必须存在一个顶点顺序，使得某种有序模式不出现，其中模式本质上是一个有序子图。这些模式刻画已被研究数十年，但近期有工作试图更系统地理解它们。本文关注该主题核心的一个简单问题：给定一个大小为 $n$ 的有序图，检测一个固定大小为 $k$ 的模式是否存在的最快速度是多少？遵循图类识别的文献，我们首先寻找可在线性时间内检测的模式。我们证明，在包含许多有趣图类（如区间图、弦图、分割图、二部图及可比性图）的三顶点模式中，几乎全部属于此类。然后，从细粒度复杂度的角度，我们给出了该问题的条件性下界。特别地，我们证明对于一大类四顶点模式，不太可能存在次二次算法。最后，我们为模式定义了一个参数——合并宽度（merge-width），并证明对于合并宽度为 $t$ 的模式，可以在 $O(n^{ct})$（其中 $c$ 为常数）时间内解决问题。作为推论，我们发现检测外平面模式及其他模式类别可在不依赖于模式大小的时间内完成。