  A graph algorithm is truly subquadratic if it runs in ${\cal O}(m^b)$ time on
connected $m$-edge graphs, for some positive $b < 2$. Roditty and Vassilevska
Williams (STOC'13) proved that under plausible complexity assumptions, there is
no truly subquadratic algorithm for computing the diameter of general graphs.
In this work, we present positive and negative results on the existence of such
algorithms for computing the diameter on some special graph classes.
Specifically, three vertices in a graph form an asteroidal triple (AT) if
between any two of them there exists a path that avoids the closed
neighbourhood of the third one. We call a graph AT-free if it does not contain
an AT. We first prove that for all $m$-edge AT-free graphs, one can compute all
the eccentricities in truly subquadratic ${\cal O}(m^{3/2})$ time. Then, we
extend our study to several subclasses of chordal graphs -- all of them
generalizing interval graphs in various ways --, as an attempt to understand
which of the properties of AT-free graphs, or natural generalizations of the
latter, can help in the design of fast algorithms for the diameter problem on
broader graph classes. For instance, for all chordal graphs with a dominating
shortest path, there is a linear-time algorithm for computing a diametral pair
if the diameter is at least four. However, already for split graphs with a
dominating edge, under plausible complexity assumptions, there is no truly
subquadratic algorithm for deciding whether the diameter is either $2$ or $3$.
