Hereditary Nordhaus–Gaddum graphs

Abstract

Nordhaus and Gaddum proved in 1956 that the sum of the chromatic number $\chi$ of a graph $G$ and its complement is at most $\left|G\right|+1$. 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 $\chi \left(H\right)+\chi \left(\overline{H}\right)\ge \left|H\right|$. 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.