A split graph is a graph whose vertices can be partitioned into a clique and an independent vertex set.
Equivalently, it is a chordal graph whose graph complement is also chordal (Royle 2000). Royle (2000) also proved that there is a one-one correspondence between the split graphs on vertices and the minimal covers of a set of size .
Classes of graphs that are split include complete , empty , star, and sun graphs.
Since all chordal graphs are perfect, so too are all split graphs.
Let be the degree sequence of a graph on vertices, and let be the largest value of such that . Then the graph is a split graph iff
Furthermore, for a graph satisfying this condition, the vertices corresponding to the first degrees in the degree sequence correspond to a maximum clique and the remainder to an independent vertex set (Golumbic 1980, Hammer and Simeone 1981).
A graph is a split graph iff it does not contain any of the graphs , , and as a forbidden induced subgraph, where is a cycle graph, is a graph complement, and is the 2-ladder rung graph (Mukhopadhyay et al. 2023).
The numbers of simple split graphs on , 2, ... vertices are given by 1, 2, 4, 9, 21, 56, 164, 557, 2223, 10766, 64956, 501696, ... (OEIS A048194).