The intensively studied Diameter problem is to find the diameter of a given connected graph. We investigate, for the first time in a structured manner, the complexity of Diameter for H-free graphs, that is, graphs that do not contain a fixed graph H as an induced subgraph. We first show that if H is not a linear forest with small components, then Diameter cannot be solved in subquadratic time for H-free graphs under SETH. For some small linear forests, we do show linear-time algorithms for solving Diameter. For other linear forests H, we make progress towards linear-time algorithms by considering specific diameter values. If H is a linear forest, the maximum value of the diameter of any graph in a connected H-free graph class is some constant dmax dependent only on H. We give linear-time algorithms for deciding if a connected H-free graph has diameter dmax, for several linear forests H. In contrast, for one such linear forest H, Diameter cannot be solved in subquadratic time for H-free graphs under SETH. Moreover, we even show that, for several other linear forests H, one cannot decide in subquadratic time if a connected H-free graph has diameter dmax under SETH.

翻译：Diameter问题旨在求解给定连通图的直径，该问题已得到深入研究。本文首次以结构化方式探讨了无H图中Diameter问题的计算复杂性，即不包含固定图H作为导出子图的图类。我们首先证明，若H不是具有小组件的线性森林，则在SETH假设下，无H图中的Diameter问题无法在次二次时间内求解。对于某些小型线性森林，我们确实给出了求解Diameter的线性时间算法。对于其他线性森林H，我们通过考虑特定直径值来推进线性时间算法的研究。若H为线性森林，则连通无H图类中任意图直径的最大值为仅依赖于H的常数dmax。针对多个线性森林H，我们给出了在连通无H图中判定直径是否为dmax的线性时间算法。相比之下，对于其中一种线性森林H，在SETH假设下，无H图中的Diameter问题无法在次二次时间内求解。此外，我们进一步证明，对于其他几种线性森林H，在SETH假设下，无法在次二次时间内判定连通无H图的直径是否为dmax。