A triple of independent vertices of a graph G is
called an asteroidal triple} (AT) if between
any two of them there exists a path in G that does
not intersect the neighborhood of the third. In this
paper we consider different classes of graphs that
are related to AT-free graphs. We start by examining
AT-free line graphs, give a characterization of
them, and apply this for showing that all connected
AT-free line graphs are traceable. In the second
part we consider line graphs of AT-free graphs. Here
we prove that every AT-free graph contains an
edge-dominating trail, and that, consequently, every
line graph of an AT-free graph is
traceable. Moreover, we give an algorithm to find
such an edge-dominating trail. In the third part of
the paper we consider claw-free AT-free graphs and
show a couple of Hamiltonian properties for them,
using the {sc Ryj{a} c}ek} closure. In the last
section we give a characterization of all AT-free
graphs with maximum degree at most 3.
