  We give characterizations of the structure and degree sequences of hereditary
unigraphs, those graphs for which every induced subgraph is the unique
realization of its degree sequence. The class of hereditary unigraphs properly
contains the threshold and matrogenic graphs, and the characterizations
presented here naturally generalize those known for these other classes of
graphs.
  The degree sequence characterization of hereditary unigraphs makes use of the
list of values $k$ for which the $k$th Erd\H{o}s--Gallai inequality holds with
equality for a graphic sequence. Using the canonical decomposition of
Tyshkevich, we show how this list describes structure common among all
realizations of an arbitrary graphic sequence.
