  The neighborhood degree list (NDL) is a graph invariant that refines
information given by the degree sequence and joint degree matrix of a graph and
is useful in distinguishing graphs having the same degree sequence. We show
that the space of realizations of an NDL is connected via a switching
operation. We then determine the NDLs that have a unique realization by a
labeled graph; the characterization ties these NDLs and their realizations to
the threshold graphs and difference graphs.

Combinatorics