🤖 AI Summary
This paper systematically investigates the computational complexity boundary of diameter computation on $H$-free graphs—graphs excluding $H$ as an induced subgraph. Focusing on the case where $H$ is a linear forest, it establishes the first precise dichotomy between subquadratic-time solvability and hardness: for several classes of linear forests, it provides linear-time algorithms to either compute the exact diameter or determine its extremal value $d_{max}$; conversely, under the Strong Exponential Time Hypothesis (SETH), it proves that diameter computation admits no subquadratic-time algorithm for other linear forests. Methodologically, the work integrates structural graph decomposition, induced-subgraph exclusion reasoning, and conditional lower-bound analysis. Its core contribution is identifying the structural parameters of $H$—particularly the number and lengths of connected components in a linear forest—that govern the computational complexity of diameter computation. This yields the first fine-grained classification, bridging the gap from “hard” to “linear-time solvable,” thereby resolving a longstanding theoretical gap in the study of diameter problems on hereditary graph classes.
📝 Abstract
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.