The independence complex of a simple graph 
is the abstract
simplicial complex
 |
It includes the empty set. Its maximal faces are the maximal independent vertex sets of 
, and for a nonempty graph its dimension
is 
,
where 
is the independence number.
Equivalently, it is the clique complex of the graph complement, so it is a flag
complex. For a complete graph 
, it consists of 
isolated points. For an empty graph
on 
vertices, it is an 
-simplex. For 
, it is a five-edge cycle, hence homeomorphic to a circle.
Kim (2022) proves that a graph is a ternary graph iff the independence complex of every nonempty induced subgraph is contractible or homotopy equivalent to a sphere.
