The directed clique number of a finite oriented graph 
with at least one graph vertex is
 |
where 
ranges over total orders of the vertex
set and 
is the undirected graph joining 
and 
whenever 
and 
contains the arc 
. Thus 
records backward arcs,
and 
denotes its ordinary clique number (Aboulker et
al. 2026).
An acyclic digraph has directed clique number 1, while a cyclic triple has value 2. The directed clique number is at most the dichromatic number. For a tournament, this invariant generally differs from the clique number of the underlying complete graph.
Aboulker et al. (2026) note that this definition already occurs in the thesis of Kim (2013). Subsequent work includes the proof by Aubian
and Coulomb (2026) that deciding 
is NP-complete
for every fixed integer 
when 
is a tournament.
