A cograph (or "complement-reducible graph") is simple graph defined by the criteria
1. 
is a cograph,
2. If 
is a cograph, then so is its graph
complement, and
3. If 
and 
are cographs, then so is their graph
union 
(Brandstadt et al. 1999).
Note that cographs have been discovered independently many times since the 1970s, so no particular definition or terminology should be considered standard. Brandstadt et al. (1999) contains references for many of the independent discoveries/definitions/characterizations of cographs.
A graph 
is a cograph if any of the following
equivalent conditions holds:
1. 
can be constructed from isolated vertices
by disjoint union and graph join operations.
2. 
is the disjoint union of distance-hereditary
graphs with diameter at most 2.
3. In every induced subgraph 
of 
,
the intersection of any maximal clique and any
maximum independent vertex set contains
precisely one vertex.
4. Every nontrivial subgraph of 
has at least one pair of twins (i.e., two vertices with the
same neighborhoods).
5. The graph complement of every nontrivial connected subgraph of 
is disconnected.
6. Every connected subgraph of 
has diameter at most 2.
7. 
does not contain the path
graph 
as an induced subgraph.
The numbers of cographs on 
,
2, ... nodes are 1, 2, 4, 10, 24, 66, 180, 522, 1532, ... (OEIS A000084).
Brandstadt et al. (1999, definition 1.5) note that a graph is a cograph if
its modular decomposition tree contains only parallel and series nodes. More specifically
and explicitly, the counts of cographs on 
nodes are the same as the counts of series-parallel
networks with 
unlabeled edges, as noted by Weisstein (2003ab) and proved by Sloane. The first cographs
on 
to 5 nodes are illustrated above.
For 
, the number of cographs is always
even.
