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：广泛研究的直径问题是指找出给定连通图的直径。我们首次以结构化的方式系统研究了H-free图（即不包含固定图H作为诱导子图的图）中直径问题的复杂性。我们首先证明：若H不是由小分支构成的线性森林，则在SETH假设下，对于H-free图，直径问题无法在次二次时间内求解。对于某些小线性森林，我们确实给出了求解直径的线性时间算法。对于其他线性森林H，我们通过考虑特定直径值，朝着线性时间算法取得进展。若H是线性森林，则连通H-free图类中任意图的最大直径是一个仅依赖于H的常数dmax。针对若干线性森林H，我们给出了判定连通H-free图直径是否为dmax的线性时间算法。相比之下，对于某个此类线性森林H，在SETH假设下，H-free图的直径问题仍无法在次二次时间内求解。此外，我们甚至证明：对于其他若干线性森林H，在SETH假设下，无法在次二次时间内判定连通H-free图直径是否为dmax。