  The extended Shi and Catalan arrangements are well investigated arrangements.
In this paper, we prove that the cone of the extended Catalan arrangement of
type A is always hereditarily free, while we determine the dimension in which
the cone of the extended Shi arrangement of type A is hereditarily free. For
this purpose, using digraphs, we define a class of arrangements which is closed
under restriction, and which contains the extended Shi and Catalan
arrangements. We also characterize the freeness for the cone of this
arrangement by graphical conditions.
