A split graph is a finite simple graph whose vertices can be partitioned into a clique and an independent vertex set, with either part allowed to be empty.
Equivalently, it is a chordal graph whose graph complement is also chordal (Royle 2000). Royle
(2000) also proved that there is a one-to-one
correspondence between split graphs on 
vertices up to graph
isomorphism and the minimal covers of an 
-element set up
to permutations of its elements.
Classes of graphs that are split include complete 
, empty 
, star,
and sun graphs formed from a clique
and an independent vertex set.
Since all chordal graphs are perfect, so too are all split graphs.
Let 
be the degree sequence of a finite simple
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 up to graph isomorphism on 
, 2, ... vertices
are given by 1, 2, 4, 9, 21, 56, 164, 557, 2223, 10766, 64956, 501696, ... (OEIS
A048194).
Threshold graphs form a subclass of split graphs characterized by additionally excluding 
as a vertex-induced
subgraph. Equivalently, a threshold graph
can be built by successively adding an isolated vertex
or a graph vertex adjacent to every existing graph
vertex (Li and Liu 2026, p. 2).
For each fixed positive integer 
, a graph intersection
of 
split graphs has chromatic
number at most 
,
where 
is its clique number. More general Boolean
combinations of graphs formed from 
split graphs have chromatic
number at most 
(Adenwalla et al. 2026, pp. 29-30).
