  Nordhaus and Gaddum proved in 1956 that the sum of the chromatic number
$\chi$ of a graph $G$ and its complement is at most $|G|+1$. The
Nordhaus-Gaddum graphs are the class of graphs satisfying this inequality with
equality, and are well-understood. In this paper we consider a hereditary
generalization: graphs $G$ for which all induced subgraphs $H$ of $G$ satisfy
$\chi(H) + \chi(\overline{H}) \le |H|$. We characterize the forbidden induced
subgraphs of this class and find its intersection with a number of common
classes, including line graphs. We also discuss $\chi$-boundedness and
algorithmic results.
