A flag complex is an abstract simplicial complex 
in which every finite set of vertices that are pairwise
joined by edges spans a simplex.
Equivalently, every finite clique in the 1-skeleton
of 
is the vertex set of a simplex
of 
. This is an intrinsic condition on the
simplices belonging to 
.
If 
denotes the 1-skeleton,
then 
is a flag complex iff 
, where 
is the clique complex construction.
Thus a flag complex can be recovered as the clique
complex of its 1-skeleton, while the clique
complex of every graph is a flag complex (Jonsson 2008,
Kahle 2009).
The complex consisting of a simplex and all its faces is a flag complex. A simplicial complex of dimension at most one is a flag complex iff its 1-skeleton is a triangle-free graph. The boundary of a triangle is not a flag complex, since its three vertices form a clique but do not span a 2-simplex.
