Simplicial Flag Complexes

Perhaps the simplest situation of an abstract simplicial complex derived from combinatorial data is that of a flag complex. For a graph G and a subset S CV (G) of its vertices, we let G[ S ] denote the corresponding induced graph. 

Definition 9.1. Given an arbitrary graph G, we let C1(G) denote the abstract simplicial complex whose set of vertices is V(G) and whose simplices are all subsets S CV (G) such that G[iS ] is a complete graph. 

The abstract simplicial complex Cl (G) has various names it is called a flag complex in algebraic topology, while it is called a clique complex in combinatorics, prompted by the fact that clique is another term used in graph theory for complete subgraphs. 

Since independent sets of G are the same as the cliques of G, we see that Ind (G) is isomorphic to Cl (G) as an abstract simplicial complex. 

---

The prodsimplicial flag construction allows one to specify a prodsimplicial complex by a relatively compact set of data. The reader should note that while a simplicial complex is always also a prodsimplicial complex, a simplicial flag complex is usually not a prodsimplicial flag complex. An example of that is provided by a hollow square. 

Proposition 18.1 comes in handy when we need to show that certain prod-simplicial complexes caimot be represented as Horn complexes, since it imposes the rather rigid restriction of being a prodsimplicial flag complex. 

---