An induced subcomplex of a simplicial complex 
on a subset 
of its vertices
is the subcomplex consisting of all simplices of 
whose vertices
belong to 
.
It is also called the full subcomplex on 
and is denoted ![K[W]](/images/equations/InducedSubcomplex/Inline6.svg)
.
For example, three vertices of a simplicial complex induce a filled triangle precisely when that triangle is a simplex of the original complex. If only its three edges are present, the induced subcomplex is a circle. An induced subcomplex therefore records higher-dimensional faces as well as the induced subgraph of the edge graph.
The inclusion maps of induced subcomplexes are used to define a tight triangulation.
