Split graphs and Block Representations

In this paper, we study split graphs and related classes of graphs from the perspective of their sequence of vertex degrees and an associated lattice under majorization. Following the work of Merris in 2003, we define blocks $[\alpha(\pi)|\beta(\pi)]$, where $\pi$ is the degree sequence of a graph, and $\alpha(\pi)$ and $\beta(\pi)$ are sequences arising from $\pi$. We use the block representation $[\alpha(\pi)|\beta(\pi)]$ to characterize membership in each of the following classes: unbalanced split graphs, balanced split graphs, pseudo-split graphs, and three kinds of Nordhaus-Gaddum graphs (defined by Collins and Trenk in 2013). As in Merris' work, we form a poset under the relation majorization in which the elements are the blocks $[\alpha(\pi)|\beta(\pi)]$ representing split graphs with a fixed number of edges. We partition this poset in several interesting ways using what we call amphoras, and prove upward and downward closure results for blocks arising from different families of graphs. Finally, we show that the poset becomes a lattice when a maximum and minimum element are added, and we prove properties of the meet and join of two blocks.