Split graphs and Block Representations

TL;DR

This paper introduces a novel lattice framework for split graphs using block representations derived from degree sequences, enabling new characterizations and structural insights into various graph classes.

Contribution

It defines block representations for split graphs and related classes, establishing a poset and lattice structure based on majorization of degree sequences.

Findings

Characterization of split graph classes via block representations

Formation of a poset and lattice structure under majorization

Closure properties and structural relations among graph classes

Abstract

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.