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).
