Split graphs and Nordhaus–Gaddum graphs

A graph GG is an NG-graph if χ(G)+χ(G¯)=|V(G)|+1. We characterize NG-graphs solely from degree sequences leading to a linear-time recognition algorithm. We also explore the connections between NG-graphs and split graphs. There are three types of NG-graphs and split graphs can also be divided naturally into two categories, balanced and unbalanced. We characterize each of these five classes by degree sequence. We construct bijections between classes of NG-graphs and balanced and unbalanced split graphs which, together with the known formula for the number of split graphs on nn vertices, allows us to compute the sizes of each of these classes. Finally, we provide a bijection between unbalanced split graphs on nn vertices and split graphs on n−1n−1 or fewer vertices providing evidence for our conjecture that the rapid growth in the number of split graphs comes from the balanced split graphs.