A theorem of Ding, Oporowski, Oxley, and Vertigan implies that any sufficiently large twin-free graph contains a large matching, a co-matching, or a half-graph as a semi-induced subgraph. The sizes of these unavoidable patterns are measured by the matching index, co-matching index, and half-graph index of a graph. Consequently, graph classes can be organized into the eight classes determined by which of the three indices are bounded. We completely classify the parameterized complexity of Independent Set, Clique, and Dominating Set across all eight of these classes. For this purpose, we first derive multiple tractability and hardness results from the existing literature, and then proceed to fill the identified gaps. Among our novel results, we show that Independent Set is fixed-parameter tractable on every graph class where the half-graph and co-matching indices are simultaneously bounded. Conversely, we construct a graph class with bounded half-graph index (but unbounded co-matching index), for which the problem is W[1]-hard. For the W[1]-hard cases of our classification, we review the state of approximation algorithms. Here, we contribute an approximation algorithm for Independent Set on classes of bounded half-graph index.

翻译：Ding、Oporowski、Oxley和Vertigan的一个定理表明，任何足够大的无孪生图都包含一个大的匹配、一个共匹配或一个半图作为半诱导子图。这些不可避免模式的大小由图的匹配指数、共匹配指数和半图指数来衡量。因此，图类可以根据这三个指数中哪些有界划分为八类。我们完全分类了独立集、团和支配集在这八类图上的参数化复杂性。为此，我们首先从现有文献中推导出多个易处理性和困难性结果，然后填补已识别的空白。在我们的新结果中，我们证明了在同时具有有界半图指数和共匹配指数的每个图类上，独立集是固定参数可处理的。相反，我们构造了一个具有有界半图指数（但无界共匹配指数）的图类，使得该问题在该类上是W[1]-困难的。对于我们分类中的W[1]-困难情形，我们回顾了近似算法的现状。在此，我们为有界半图指数图类上的独立集问题贡献了一个近似算法。