  The local spectrum of a vertex set in a graph has been proven to be very
useful to study some of its metric properties. It also has applications in the
area of pseudo-distance-regularity around a set and can be used to obtain
quasi-spectral characterizations of completely (pseudo-)regular codes. In this
paper we study the relation between the local spectrum of a vertex set and the
local spectrum of each of its subconstituents. Moreover, we obtain a new
characterization for completely pseudo-regular codes, and consequently for
completely regular codes, in terms of the relation between the local spectrum
of an extremal set of vertices and the local spectrum of its antipodal set. We
also present a new proof of the version of the Spectral Excess Theorem for
extremal sets of vertices.
