  A graph with degree sequence $\pi$ is a \emph{unigraph} if it is isomorphic
to every graph that has degree sequence $\pi$. The class of unigraphs is not
hereditary and in this paper we study the related hereditary class HCU, the
hereditary closure of unigraphs, consisting of all graphs induced in a
unigraph. We characterize the class HCU in multiple ways making use of the
tools of a decomposition due to Tyshkevich and a partial order on degree
sequences due to Rao. We also provide a new characterization of the class that
consists of unigraphs for which all induced subgraphs are also unigraphs.
