Hereditary Nordhaus–Gaddum graphs

Nordhaus and Gaddum proved in 1956 that the sum of the chromatic number χ 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 χ ( H ) + χ ( H ¯ ) ≥ | 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 χ -boundedness and algorithmic results.